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ELEMENTS  OF 
APPLIED  MATHEMATICS 


BY 
HERBERT  E.  COBB 

PROFESSOR  OF  MATHEMATICS,   LEWIS   INSTITUTE,  CHICAQO 


GINN  AND  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON 


COPYRIGHT,  1911 
By  HERBERT  E.  COBB 


ALL  RIGHTS   RESERVED 
712.9 


Vf)t  iStbtngum  gre« 

GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


PREFACE 


This  book  of  problems  is  the  result  of  four  years'  experimen- 
tation in  the  endeavor  to  make  the  instruction  in  mathematics 
of  real  service  in  the  training  of  pupils  for  their  future  work. 
There  is  at  the  present  time  a  widespread  belief  among  teach- 
ers that  the  formal,  abstract,  and  purely  theoretical  portions 
of  algebra  and  geometry  have  been  unduly  emphasized.  More- 
over, it  has  been  felt  that  mathematics  is  not  a  series  of  dis- 
crete subjects,  each  in  turn  to  be  studied  and  dropped  without 
reference  to  the  others  or  to  the  mathematical  problems  that 
arise  in  the  shops  and  laboratories.  Hence  we  have  attempted 
to  relate  arithmetic,  algebra,  geometry,  and  trigonometry  closely 
to  each  other,  and  to  connect  all  our  mathematics  with  the  work 
in  the  shops  and  laboratories.  This  has  been  done  largely  by 
lists  of  problems  based  on  the  preceding  work  in  mathematics 
and  on  the  work  in  the  shops  and  laboratories,  and  by  simple 
experiments  and  exercises  in  the  mathematics  classrooms,  where 
the  pupil  by  measuring  and  weighing  secures  his  own  data  for 
numerical  computations  and  geometrical  constructions. 

In  high  schools  where  it  is  possible  for  the  teachers  to  depart 
from  traditional  methods,  although  they  must  hold  to  a  year 
of  algebra  and  a  year  of  geometry,  this  book  of  problems  can 
be  used  to  make  a  beginning  in  the  unification  of  mathematics, 
and  to  make  a  test  of  work  in  applied  problems.  In  the  first 
year  in  algebra  the  problems  in  Chapters  I -VII  can  be  used 
to  replace  much  of  the  abstract,  formal,  and  lifeless  mate- 
rial of  the  ordinary  course.  These  problems  afford  a  much- 
needed  drill  in  arithmetical  computation,  prepare  the  way  for 
geometry,  and  awaken  the  interest  of  the  pupils  in  the  affairs 


iv  APPLIED  MATHEMATICS 

of  daily  life.  By  placing  less  emphasis  on  the  formal  side  of 
geometry  it  is  possible  to  make  the  pupil's  knowledge  of  alge- 
bra a  valuable  asset  in  solving  geometrical  problems,  and  to 
give  him  a  working  knowledge  of  angle  functions  and  log- 
arithms. Chapters  IX,  X,  and  XII  furnish  the  material  for 
this  year's  work.  The  problems  of  the  remaining  chapters  can 
be  used  in  connection  with  the  study  of  advanced  algebra  and 
solid  geometry.  They  deal  with  various  phases  of  real  life,  and 
in  solving  them  the  pupil  finds  use  for  all  his  mathematics, 
his  physics,  and  his  practical  knowledge. 

For  the  increasing  number  of  intermediate  industrial  schools 
there  are  available  at  present  few  lists  of  problems  of  the  kind 
brought  together  in  this  book.  The  methods  adopted  in  the 
earlier  chapters,  which  require  the  pupil  to  obtain  his  own  data 
by  measuring  and  weighing,  are  especially  valuable  for  begin- 
ners and  boys  who  have  been  out  of  school  for  several  years. 

The  large  number  of  problems  and  exercises  permits  the 
teacher  to  select  those  that  are  best  suited  to  the  needs  of  the 
class.  In  Chapters  IX  and  XIII  many  of  the  problems  contain 
two  sets  of  numbers.  The  first  set  outside  of  the  parentheses 
may  give  an  integral  result,  while  the  second  set  may  involve 
fractions  ;  or  the  first  set  may  give  rise  to  a  quadratic  equation 
which  can  be  solved  by  factoring,  while  the  equation  of  the 
second  set  must  be  solved  by  completing  the  square. 

Each  pupil  should  have  a  triangle,  protractor,  pair  of  com- 
passes, metric  ruler,  and  a  notebook  containing  plain  and 
squared  paper.  Inexpensive  drawing  instruments  can  be  ob- 
tained, and  the  pupils  should  be  urged  to  use  them  in  making 
rough  checks  of  computations.  They  should  also  form  the 
habit  of  making  a  rough  estimate  of  the  answer,  and  noting 
if  the  result  obtained  by  computation  is  reasonable. 

In  the  preparation  of  this  book  most  of  the  works  named  in 
the  Bibliography  have  been  consulted.  The  chapter  on  squared 
paper  aims  to  emphasize  its  chief  uses,  the  representation  of 


PREFACE  V 

tables  of  values,  and  the  solution  of  problems ;  and  to  show 
that  the  graph  should  be  used  in  a  common-sense  way  in  all 
mathematical  work. 

The  cooperation  of  the  members  of  the  department  of  mathe- 
matics in  the  Lewis  Institute  in  the  work  of  preparing  and 
testing  the  material  for  this  book  has  rendered  the  task  less 
burdensome ;  acknowledgments  are  due  to  Assistant  Professor 
D.  Studley  for  the  problems  in  Chapters  XIV  and  XV;  to 
Assistant  Professor  B.  J.  Thomas  for  aid  in  Chapters  I,  VIII, 
XII,  and  XIII ;  to  Mr.  E.  H.  Lay  for  aid  in  Chapters  II  and  VI ; 
and  to  Mr.  A.  W.  Cavanaugh  for  aid  in  Chapter  IX.  Especial 
acknowledgments  are  due  to  Professor  P.  B.  Woodworth,  head 
of  the  department  of  physics,  Lewis  Institute,  for  his  helpful 
cooperation  with  the  work  of  the  mathematics  department. 


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CONTENTS 

CHAPTER  FAO^ 

I.   Measurement  and  Approximate  Number  ...  1 

II.   Vernier  and  Micrometer  Calipers 9 

III.  Work  and  Power 16 

IV.  Levers  and  Beams 27 

V.   Specific  Gravity 42 

VI.   Geometrical     Constructions    with    Algebraic 

Applications 52 

VII.   The  Use  of  Squared  Paper 65 

VIII.   Functionality;  Maximum  and  Minimum  Values  91 
IX.   Exercises    for   Algebraic    Solution   in   Plane 

Geometry 97 

X.   Common  Logarithms 120 

XL   The  Slide  Rule 128 

XII.   Angle  Functions 134 

XIII.  Geometrical  Exercises  for  Advanced  Algebra  153 

XIV.  Variation 164 

XV.   Exercises  in  Solid  Geometry 177 

XVI.   Heat 195 

XVIL   Electricity 212 

XVIII.   Logarithmic  Paper 243 

TABLES 258 

BIBLIOGKAPHY 261 

FOUR-PLACE  LOGARITHMS 265 

INDEX 273 


I 


APPLIED  MATHEMATICS 

CHAPTER  I 

MEASUREMENT  AND  APPROXIMATE  NUMBER 

Exercise.  Make  a  sketch  of  the  whitewood  block  that  has 
been  given  you ;  measure  its  length,  breadth,  and  thick  n-ess  in 
millimeters  and  write  the  dimensions  on  the  sketch.  Find  the 
volume  of  the  block.  Have  you  found  the  exact  volume  ? 
Were  your  measurements  absolutely  correct  ? 

1.  Errors.  In  making  measurements  of  any  kind  there  are 
always  errors.  We  do  not  know  whether  or  not  the  foot  rule, 
the  meter  stick,  or  the  100-foot  steel  tape  we  are  using  is  abso- 
lutely exact  in  length  and  graduation.  Hence  one  source  of 
error  lies  in  the  instruments  we  use.  Another  source  of  error 
is  the  inability  to  make  correct  readings.  When  you  attempt 
to  measure  the  length  of  a  whitewood  block,  you  will  probably 
find  that  the  corners  are  rather  blunt,  making  it  impossible  to 
set  a  division  of  the  scale  exactly  on  the  corner.  Moreover,  it 
is  seldom  that  the  end  of  the  line  you  are  measuring  appears 
to  coincide  exactly  with  a  division  of  the  scale.  If  you  are  using 
a  scale  graduated  to  millimeters  and  record  your  measurements 
only  to  millimeters,  then  a  length  is  neglected  if  it  is  less  than 
half  a  millimeter,  and  called  one  millimeter  if  it  is  greater  than 
half  a  millimeter. 

To  make  a  reading  as  correct  as  possible,  be  sure  that  the  eye 
is  placed  directly  over  the  division  of  the  scale  at  which  the 
reading  is  tnade.    Note  if  the  end  of  the  scale  is  perfect. 

1 


2  APPLIED  MATHEMATICS 

2.  Significant  figures.  A  digit  is  one  of  the  ten  figures  used 
in  number  expressions.  A  significant  figure  is  a  digit  used  to 
express  the  amount  which  enters  the  number  in  that  particular 
place  which  the  digit  occupies.  All  figures  other  than  zero  are 
significant.  A  zero  may  or  may  not  be  significant.  It  is  sig- 
nificant if  written  to  show  that  the  quantity  in  that  place  is 
nearer  to  zero  than  to  any  other  digit,  but  a  zero  written  merely 
to  locate  the  decimal  point  is  not  significant.  A  zero  inclosed 
by  other  digits  is  significant,  while  a  final  zero  may  or  may  not 
be  significant. 

For  example,  in  the  number  0.0021  the  zeros  are  not  signifi- 
cant. In  the  number  .0506  the  first  zero  is  not  significant,  while 
the  zero  inclosed  by  the  5  and  6  is  significant.  If  in  a  measure- 
ment a  result  written  as  56.70  means  that  it  is  nearer  56.70 
than  56.69  or  56.71,  the  zero  is  significant.  In  saying  that  a 
house  cost  about  $6700,  the  final  zeros  are  not  significant  be- 
cause they  merely  take  the  place  of  other  figures  whose  value 
we  do  not  know  or  do  not  care  to  express. 

3.  Exact  numbers.  In  making  computations  with  exact 
numbers,  multiplications  and  divisions  are  done  in  fidl,  accord- 
ing to  methods  which  are  familiar  to  all  students. 

4.  Approximate  numbers.  In  practical  calculations  most  of 
the  numbers  used  are  not  exact  but  are  approximate  numbers. 
They  are  obtained  by  measuring,  weighing,  and  other  similar 
processes.  Such  numbers  cannot  be  exact,  for  instruments  are 
not  perfect  and  the  sense  of  vision  does  not  act  with  absolute 
precision.  If  the  length  of  a  rectangular  piece  of  paper  were 
measured  and  found  to  be  614  mm.,  the  6  and  the  1  woidd 
very  likely  be  exact,  but  the  4  would  be  doubtful. 

5.  Multiplication  of  approximate  numbers.  This  contracted 
method  of  multiplication  gives  the  proper  number  of  significant 
figures  in  the  product  with  no  waste  of  labor.  Moreover,  by 
omitting  the  doubtful  figures  it  avoids  an  appearance  of  great 
accuracy  in  the  result,  which  is  not  warranted  by  the  data. 


MEASUREMENT  AND  APPROXIMATE  NUMBER        3 

Exercise.  Measure  the  length  and  width  of  a  rectangular 
piece  of  paper  and  find  its  area. 

Suppose  the  length  is  614  mm.  and  the  width  is  237  mm. 
Let  us  proceed  to  find  the  area  of  the  piece  of  paper,  marking 
the  doubtful  figures  throughout  the  work. 

237 

614 

948 
237 
1422 
145518 

The  final  three  figures  in  the  product  are  doubtful  and  may 
as  well  be  replaced  by  zeros.  Hence  the  area  is  approximately 
145,000  sq.  mm.,  or,  as  we  sometimes  say,  about  145,000 
sq.  mm.  Since  many  calculations  are  of  this  kind,  it  is  a  waste 
of  time  to  carry  out  the  operations  in  full.  It  is  desirable  to 
use  methods  which  will  omit  the  doubtful  figures  and  retain 
only  those  which  are  certain. 

Problem.    Multiply  24.6  by  3.25. 

First  step  Second  step  Third  step 

24.6  24.^  2^.^ 

3.25  3.25  3.25 

738  738  738 

49  49 

12 

79.9 

First  step.  Start  with  3  at  the  left  in  the  multiplier  and 
write  the  partial  product  as  shown. 

Second  step.  Cut  off  the  6  in  the  multiplicand  and  multiply 
by  2.  Twice  6  (mentally)  are  12  (1.2),  which  gives  1  to  add. 
Twice  4  are  8,  and  1  to  add  makes  9.    Twice  2  are  4. 

Third  step.  Cut  off  the  4  in  the  multiplicand  and  multiply 
by  5.  5  times  4  (mentally)  are  20  (2.0),  which  gives  2  to  add. 
5  times  2  are  10,  and  2  to  add  makes  12. 


4  APPLIED  MATHEMATICS 

Fourth  step.    Add  the  partial  products. 

Fifth  step.  Place  the  decimal  point  by  considering  the  num 
ber  of  integral  figures  which  the  product  should  contain.  Thii 
may  usually  be  done  by  making  a  rough  estimate  mentally- 
In  this  case  we  see  that  3  times  24  are  72,  and  by  estimating 
the  amount  to  be  brought  up  from  the  remaining  parts  we  see 
that  the  product  is  more  than  75.  Hence  there  are  two  inte- 
gral figures  to  b6  pointed  off. 

Problem.    Multiply  84.6  by  4.25. 

Third  step 


First  step 

Second  step 

84.^ 

8^.^ 

4.25 

4.25 

338 

338 

17 

4.25 
338 
17 
4 
359 
In  this  case  6  is  cut  off  before  multiplying  by  4  in  order  to 
keep  the  product  to  three  figures.    The  two  given  numbers  are 
doubtful  in  the  third  figure,  and  usually  this  makes  the  product 
doubtful  in  the  third  figure. 

Problem.    Find  the  product  of  tt  x  3.784  x  460.2. 

Solution.          ^.%i%  UW 

3.784  460.2 

9426  4756 

2199  713 

251  ^  2 

12  *                5471 


11.888 
6.  Measurements.  In  making  measurements  to  com,pute 
areas,  volum.es,  and  so  on,  all  parts  should  he  measured  with 
the  same  relative  accuracy  ;  that  is,  they  should  all  be  expressed 
with  the  same  number  of  significant  figures.  The  calculated 
parts  should  not  show  more  significant  figures  than  the  meas- 
ured parts.  Constants  like  tt  should  be  cut  to  the  same  number 
of  figures  as  the  measured  parts. 


MEASUREMENT  AND  APPROXIMATE  NUMBER        5 

EXERCISES 

1.  Find  the  area  of  the  printed  portion  of  a  page  in  your 
algebra. 

2.  Find  the  volume  of  your  algebra. 

3.  Find  the  area  of  the  top  of  your  desk. 

4.  Find  the  area  of  the  door. 

5.  Find  the  number  of  cubic  feet  of  air  in  the  room. 

6.  Find  the  area  of  one  section  of  the  blackboard. 

7.  Find  the  surface  and  volume  of  brass  cylinders  and 
prisms,  and  of  wooden  blocks. 

8.  Find  the  area  of  the  athletic  field. 

9.  Find  the  area  of  the  ground  covered  by  the  school  build- 
ings and  also  the  area  of  some  of  the  halls  and  recitation  rooms. 
Compare  your  results  with  computations  made  from  the  plans 
of  the  buildings,  if  they  are  accessible. 

7.  Division  of  approximate  numbers.  In  dividing  one  ap- 
proximate number  by  another,  we  shorten  the  work  by  cutting 
off  figures  in  the  divisor  instead  of  adding  zeros  in  the  dividend. 
The  principles  of  contracted  multiplication  are  used  in  the 
multiplication  of  the  divisor  by  the  figures  of  the  quotient. 
No  attention  is  paid  to  the  decimal  point  in  the  dividend  or 
divisor  till  the  quotient  has  been  obtained.  In  checking  multi- 
ply the  quotient  by  the  divisor.    (Why  ?) 

Problems.   1.  Divide  83.62  by  3.194. 

3J^  83.62  [2618  Check 

6388  2^.;? 


1974  3.194 

1916  7854 

58  262 

32  235 

26  10 

25  83.61 
1 


APPLIED  MATHEMATICS 


2.  Divide  41.684  by  98.247. 

?^.^^J  141.684 1.42428    - 

Check 

39299 
2385 

•  98.247 

1965 

38185 

420 

3394 

393 

27 

85 
17 

20      - 

3 

41.684 


The  decimal  point  in  the  quotient  can  usually  be  placed 
quite  easily  by  considering  the  number  of  integral  figures  in 
the  divisor  and  dividend.  In  the  first  problem  we  see  that  3  is 
contained  in  83  about  26  times ;  in  the  second  problem  98  is 
contained  in  41  about  .4  times. 


PROBLEMS 

Check  the  results  obtained : 


1.  2.142  X  3.152. 

2.  78.14  X  1.314. 

3.  6.718  X  86.42. 

4.  3.142  X  .7854. 

5.  (1.4142)2. 

6.  (1.732)2. 

7.  (3.142)2. 

8.  (5.164)8. 

9.  (.6462)^ 


10.  86.66^41.37. 

11.  316.4 -- 18.74. 

12.  .916 -.314. 
14.16  X  5.873 

8.614 
14.  3.142  X  (1.666)2. 
36.5  X  192 


15 


16. 


4.12  X  6.33 

4  X  3.142  X  (6.Q23)^ 


17.  An  iron  bar  is  9.21  in.  by  2.43  in.  by  1.12  in.    Find  its 
weight  if  1  cu.  in.  of  iron  weighs  .261  lb. 

18.  Find  the  weight  of  a  block  of  oak  5.62  in.  by  3.92  in.  by 
3.15  in.  if  1  cu.  in.  of  oak  weighs  .0422  lb. 


MEASUREMENT  AND  APPROXIMATE  NUMBER        7 

19.  Find  the  weight  of  an  iron  plate  125  in.  long,  86.2  in. 
wide,  and  .562  in.  thick. 

20.  The  diameter  of  a  piston  is  16.4  in.  Find  its  area. 
(tt  =  3.14.) 

21.  The  radius  of  a  circle  is  12.67  in.  Find  its  area. 
(tt  =  3.142.) 

22.  The  diameter  of  a  steam  boiler  is  56.8  in.  What  is  its 
circumference  ? 

23.  The  area  of  a  rectangle  is  25.37  sq.  in.  Find  the  width 
if  the  length  is  11.42  in. 

24.  Wliat  is  the  length  of  a  cylinder  whose  volume  is  1627 
cu.  in.  if  the  area  of  a  cross  section  is  371.5  sq.  in.  ? 

25.  A  cylindrical  safety-valve  weight  of  cast  iron  is  15j  in. 
in  diameter  and  3^  in.  thick.  Find  its  weight  if  1  cu.  in.  of 
cast  iron  weighs  .261  lb. 

26.  A  cylindrical  safety-valve  weight  of  cast  iron  weighs 
82.5  lb.    What  is  its  diameter  if  it  is  Ij  in.  thick  ? 

27.  The  diameter  of  a  spherical  safety  valve  of  cast  iron  is 
9.3  in.    Find  its  weight. 

28.  Find  the  weight  of  a  cast-iron  pipe  28.5  in.  long  if  the 
outer  diameter  is  10.9  in.  and  the  inner  diameter  is  9.2  in. 

29.  A  cylindrical  water  tank  is  49.6  in.  long  and  its  diameter 
is  28.6  in.    Find  its  volume.    How  many  gallons  will  it  hold  ? 

30.  A  steel  shaft  is  68.8  in,  long  and  its  diameter  is  2.58  in. 
Find  its  weight  if  1  cu.  in.  of  steel  weighs  .283  lb. 

31.  Find  the  weight  of  the  water  in  a  full  cylindrical  water 
tank  12.8  ft.  in  height  and  6.32  ft.  in  diameter  if  1  cu.  ft.  of 
water  weighs  62.4  lb. 

32.  The  diameter  of  the  wheels  over  which  a  band  saw  runs 
is  3.02  ft.  and  the  distance  between  the  centers  of  the  pulleys 
is  3.58  ft.    Find  the  length  of  ithe  saw. 

33.  A  pulley  11.9  in.  in  diameter  is  making  185  revolutions  per 
minute  (r.  p.  m.).    How  fast  is  the  rim  traveling  per  minute  ? 


8  APPLIED  MATHEMATICS 

34.  A  milling  cutter  4  in.  in  diameter  is  running  150  r.  p.  m. 
What  is  the  surface  speed  in  feet  per  minute  ? 

35.  It  is  desired  to  make  a  12-in.  emery  wheel  run  at  a  speed 
of  5000  ft.  per  minute.  How  many  revolutions  per  minute  must 
it  make  ? 

36.  If  we  wish  a  milling  cutter  to  run  at  a  cutting  speed  of 
266  ft.  per  minute,  and  the  machine  can  make  only  82  r.  p.  m., 
what  must  be  the  diameter  of  the  cutter  ? 


CHAPTER  II 

VERNIER  AND  MICROMETER  CALIPERS 

8.  The  vernier  calipers  have  two  jaws  between  which  is  placed 
the  object  to  be  measured.  One  jaw  slides  on  a  bar  which  has 
scales,  on  one  side  centimeters  and  on  the  other  side  inches. 


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liiliiiiliiTiliiiiliinliiiiliiiiliiiiliiiillilililil 


Fig.  1 

The  movable  jaw  has  two  small  scales  called  verniers,  one  for 
each  main  scale. 

Write  the  following  questions  and  their  answers  in  your 
notebook.  Use  the  centimeter  for  the  unit  and  write  the  results 
as  decimal  fractions. 

1.  (a)  How  many  centimeters  are  marked  on  the  main 
scale  ?  (b)  Verify  by  measuring  with  the  ruler,  (c)  What  is 
the  length  of  the  smallest  division  of  the  main  scale  ? 

2.  (a)  What  is  the  length  of  the  centimeter  vernier? 
(h)  Measure  the  length  of  the  vernier  with  the  ruler,  (c)  Verify 
by  counting  the  divisions  on  the  main  scale. 

3.  (a)  Into  how  many  divisions  is  the  vernier  scale  divided  ? 
(b)  What  is  the  length  of  each  division  ? 

9 


10  APPLIED  MATHEMATICS 

4.  Bring  the  jaws  of  the  calipers  together.    At  what  point 
on  the  main  scale  does  the  first  line  of  the  vernier  fall  ? 

Make  a  drawing  of  the  vernier  and  the  scale  as  suggested 
by  Fig.  2.    Number  the  points  of  division  as  in  the  figure. 


MAIN    6CALE 
O     /     Z     3    4     S      6     7     Q     9    /O   t/     IZ   J3    /4    JS 


,\      \,\      I      I.    I       1.    t,    ,,    , 
O    /'  2    3'  4'  S    6    7    e    9  /O' 


Fig.  2 

Slide  the  vernier  of  the  calipers  along  until  0'  coincides  with  0. 

5.  (a)  Do  V  and  1  coincide  ?  (h)  What  is  the  distance 
between  1'  and  1  ?  (c)  between  2'  and  2  ?  (d)  between  3' 
and  3? 

Now  slide  the  vernier  along  until  1'  and  1  coincide. 

6.  (a)  What  is  the  distance  between  0  and  0'  ?  (b)  between 
2'  and  2  ? 

Make  2'  and  2  coincide. 

7.  What  is  the  distance  between  0  and  0'  now  ? 

8.  What  is  the  distance  between  0  and  0'  when  the  follow- 
ing points  coincide  ?  (a)  3'  and  3 ;  (^)  4'  and  4 ;  (c)  7'  and  7  ; 
{d)  9'  and  9 ;  (e)  10'  and  10 

Move  the  vernier  until  0'  coincides  with  10. 

9.  How  far  apart  are  the  jaws  ?    Check  with  the  ruler. 

10.  When  0'  coincides  with  20,  how  far  apart  are  the  jaws  ? 
Check. 

11.  When  0'  coincides  with  21,  how  far  apart  are  the  jaws  ? 

12.  What  is  the  distance  between  the  jaws  when  the  follow- 
ing points  coincide  ?  (r?)  I'and  22;  {b)  2'  and  23;  (c)  5'  and  26; 
{d)  8'  and  29;  (e)  1'  and  23;  (/)  V  and  29;  {g)  2'  and  26; 
{h)  3'  and  35. 

9.  To  measure  with  the  vernier.  Count  the  number  of  irhole 
centimeters  and  mUlim,eters  to  the  zero  line  of  the  vernier.   Then 


VERNIER  AND  MICROMETER  CALIPERS  11 

notice  which  vernier  division  coincides  with  a  scale  division  ;  the 
number  of  this  vernier  division  is  the  number  of  tenths  of  a 
millimeter. 

10.  Observe  carefully  the  following  directions  for  making 
measurements.  Unlock  the  movable  jaw  by  means  of  the  screw 
at  the  side.  Place  the  object  between  the  jaws,  press  these 
together  gently  but  firmly  with  the  fingers,  and  lock  in  position 
with  the  screw.  Care  should  be  taken  in  pressing  upon  the 
jaws  as  too  strong  a  pressure  may  injure  the  instrument.  If 
not  enough  pressure  is  applied,  the  reading  will  not  be  accurate. 

EXERCISES 

1.  Place  your  pencil  between  the  jaws  of  the  calipers  and 
measure  its  diameter. 

2.  Get  a  block  from  your  instructor  and  measure  its  dimen- 
sions. Make  a  record  of  them  together  with  the  number  of  the 
block,  and  let  the  instructor  check  the  results. 

3.  Get  a  second  block.  Make  measurements  of  the  length 
at  three  different  places  on  the  block  and  record  them.  Take 
the  average  of  the  three  readings.  Find  dimensions  in  the 
same  way.    Let  the  instructor  check  the  record. 

4.  Draw  an  indefinite  line  AB.  With  a  point  R  about  1  cm. 
from  AB  as  a  center,  and  with  a  radius  of  3  cm.  draw  a  circle 
intersecting  AB  'dt  C  and  D.  Measure  CD,  making  the  measure- 
ment with  the  pointed  ends  of  the  jaws.    Check  your  reading. 

5.  Take  a  sheet  of  squared  paper  and  fix  the  vertices  of  a 
square  centimeter  with  the  point  of  the  compasses.  Measure 
the  diagonals  and  take  the  average.    Check. 

6.  On  the  same  sheet  of  squared  paper  locate  the  vertices 
of  a  rectangle  4  cm.  by  2.5  cm.  Measure  the  diagonals  and 
check  the  results. 

7.  Apply  the  sets  of  questions  in  these  exercises  to  the  inch 
scale  and  its  vernier,  inserting  the  word  "inch"  for  "centi- 
meter "  in  your  record. 


12  APPLIED  MATHEMATICS 

8.  Measure  the  length  of  a  block  in  inches  and  in  centi- 
meters and  find'  out  the  number  of  centimeters  in  one  inch. 

9.  On  a  sheet  of  squared  paper  mark  out  a  right  triangle 
with  the  legs  3  in.  and  4  in.  respectively.  Locate  the  vertices 
with  the  point  of  the  compasses  and  measure  the  hypotenuse. 
Show  that  the  square  of  the  hypotenuse  is  equal  to  the  sum  of 
the  squares  of  the  other  two  sides. 

10.  Move  the  zero  line  of  the  vernier  opposite  1  in,  on  the 
main  scale.  Make  the  reading  in  centimeters.  Compare  the 
result  with  that  obtained  in  Exercise  8. 

11.  Find  the  volume  of  a  block  in  cubic  inches  and  also  in 
cubic  centimeters.  Check  by  changing  the  cubic  inches  into 
cubic  centimeters. 

12.  Devise  other  exercises  in  measurement. 

11.  The  micrometer  calipers.  With  the  micrometer  calipers 
the  object  to  be  measured  is  placed  between  a  revolving  rod 
called  the  screw,  and  a  fixed 
stop.  The  movable  rod  is 
turned  by  the  barrel,  which 
moves  over  a  linear  scale.  The 
edge  of  the  barrel  is  gradu- 
ated into  a  circular  scale.  _ 

,«    •,,         ,    ,  .  Fig.  3 

12.  Use  of  the  micrometer 

calipers.  Turn  the  barrel  so  that  the  screw  approaches  the  stop 
and  finally  comes  in  contact  with  it.  Now  turn  in  the  opposite 
direction  and  the  screw  moves  away  from  the  stop ;  at  the 
same  time  the  edge  of  the  barrel  moves  over  the  linear  scale, 
which  shows  the  distance  of  the  opening.  When  an  object  is 
placed  in  the  opening  between  the  stop  and  the  screw,  its 
measurement  is  obtained  by  reading  the  length  of  the  linear 
scale  exposed  to  view. 


VERNIER  AND  MICROMETER  CALIPERS  13 

EXERCISES 

Write  the  following  questions  and  their  answers  in  your 
notebook.    Express  your  results  in  decimal  fractions. 

Turn  the  barrel  until  the  entire  linear  scale  is  shown 

1.  How  many  divisions  are  marked  on  the  linear  scale  ? 

2.  Determine  the  unit  of  the  linear  scale,  whether  it  is  a 
centimeter  or  an  inch.  This  can  be  done  by  comparison  with 
the  English  and  the  metric  scales  marked  on  your  ruler. 

3.  How  long  in  inches  or  centimeters  is  the  linear  scale  ? 

4.  What  is  the  length  of  each  division  of  the  linear  scale  ? 

Turn  the  barrel  until  the  screw  conies  in  contact  with 
the  stop. 

5.  Into  how  many  divisions  is  the  circular  scale  along  the 
edge  of  the  barrel  graduated  ? 

6.  (a)  Does  the  zero  line  of  the  circular  scale  coincide  with 
the  line  of  reference  of  the  linear  scale  ? 

(i)  How  far  are  they  apart  ?  Count  the  number  of  divisions 
of  the  circular  scale  between  them.  This  is  known  as  taking 
the  zero  reading. 

Turn  the  barrel  until  the  zero  line  coincides  with  the  line 
of  reference.  Erom  this  position  turn  the  barrel  until  two 
divisions  of  the  linear  scale  have  been  passed  over. 

7.  How  many  complete  turns  were  made  ? 

Bring  the  zero  line  of  the  barrel  back  to  the  line  of  reference 
of  the  linear  scale.  Give  the  barrel  several  complete  turns  and 
count  the  number  of  divisions  passed  over  on  the  linear  scale. 
The  relation  between  the  number  of  turns  and  the  number  of 
divisions  should  be  carefully  noted. 

.8.  How  many  divisions  are  passed  over  in  (a)  two  turns? 
(V)  four  turns  ?  (<?)  six  turns  ?  {(1)  one  turn  ? 

9.  How  far  in  centimeters  or  inches  does  the  barrel  move  in 
one  complete  turn  ? 


14  APPLIED  MATHEMATICS 

Bring  the  zero  line  opposite  the  line  of  reference.  Now 
move  the  barrel  until  the  line  5  of  the  circular  scale  is  oppo- 
site the  line  of  reference. 

10.  (a)  What  part  of  a  turn  has  the  barrel  made  ? 

(b)  How  far  in  centimeters  or  inches  did  the  barrel  move  ? 

(c)  How  far  will  the  barrel  move  in  passing  over  one  division 
of  the  circular  scale  ? 

Turn  the  barrel  until  its  edge  coincides  with  the  fifth  division 
of  the  linear  scale,  and  the  zero  line  of  the  circular  scale  coin- 
cides with  the  line  of  reference. 

11.  What  is  the  length  of  the  opening  at  the  end  of  the 
screw  ?  Record  the  distance,  and  then  as  a  rough  check 
verify  by  measuring  with  a  ruler. 

With  the  barrel  in  the  same  position  as  before  (at  the  fifth 
division)  continue  to  turn  so  as  to  increase  the  opening  at  the 
end  of  the  screw.  Turn  the  barrel  until  the  seventeenth  division 
of  the  circular  scale  is  opposite  the  line  of  reference. 

12.  How  much  is  the  opening  at  the  end  of  the  screw  ? 
The  following  will  illustrate :   Suppose  the  divisions  of  the 

linear  scale  are  ^  (.025)  of  an  inch,  and  there  are  25  divisions 
on  the  circular  scale.  The  value  of  one  division  of  the  circular 
scale  will  be  ^^  x  :5V  =  -^^^  ^^-  Each  division  of  the  circular 
scale,  therefore,  measures  .001  in.  In  Exercise  12  the  distance 
for  5  linear  divisions  would  be  .025  x  5  =  .125.  This  added  to 
the  value  of  the  17  circular  divisions  gives  .125  +  .017  =  .142  in. 

for  the  reading. 

EXERCISES 

Record  the  readings  in  your  notebooks  and  give  the  work 
of  the  computations  in  full. 

1.  Measure  the  thickness  of  a  coin.  Hold  the  barrel  lightly 
so  that  it  will  slip  in  the  fingers  as  contact  occurs.  There  is 
danger  of  straining  the  screw  if  it  is  turned  up  hard.  Take 
four  readings  at  different  places  on  the  coin  and  average  the 
results. 


VERNIER  AND  MICROMETER  CALIPERS  15 

2.  Get  a  metal  solid  from  youi-  instructor  and  measure  its 
dimensions.  Take  the  average  of  three  readings.  Compute  the 
volume.    Check  by  using  an  overflow  can. 

3.  Measure  the  diameter  of  a  wire.  After  taking  a  reading 
release  the  wire  and  turn  it  about  its  axis  through  90°;  take 
a  second  reading.  If  the  two  readings  do  not  agree,  the  wire  is 
slightly  flattened  in  section.  Take  several  readings  at  different 
places  on  the  wire,  and  the  average  of  the  readings  will  proba- 
bly be  very  close  to  the  standard  diameter  of  the  wire. 

4.  Find  the  volume  of  a  shot.  Using  the  specific  gravity  of 
lead,  find  the  number  of  shot  to  the  pound. 

5.  Find  the .' thickness  of  one  of  the  pages  of  your  textbook. 
Compute  its  volume. 

6.  Devise  other  exercises  and  record  them  in  your  notebooks 


■     CHAPTER  III 

WORK  AND  POWER 

13.  Work.  When  a  man  lifts  a  bar  of  iron  or  pulls  it  along 
the  floor,  he  is  said  to  do  work  upon  it.  Evidently  it  takes 
twice  as  much  effort  to  lift  50  lb.  as  it  does  to  lift  25  lb.,  and 
five  times  as  much  effort  to  lift  a  box  5  ft.  as  it  does  to  lift  it 
1  ft.  That  is,  work  depends  upon  two  things,  —  distance  and 
pressure. 

Hence  a  foot  pound  is  taken  as  the  unit  of  work.  It  is  the 
work  done  in  raising  1  lb.  vertically  1  ft.,  or  it  is  the  pressure 
of  1  lb.  exerted  over  a  distance  of  1  ft.  in  any  direction.  If  a 
man  exerts  a  pressure  of  25  lb.  in  pushing  a  wagon  20  ft.,  he 
has  done  500  ft.  lb.  of  work. 

Foot  pounds  =  feet  x  pounds. 

14.  Illustrations.  Tie  a  string  to  a  1-lb.  weight,  attach  a 
spring  balance  and  lift  it  1,  2,  3  ft.  How  many  foot  pounds  of 
work  ?  Lower  it  1,  2,  3  ft.  How  much  work  ?  Pull  the  string 
horizontally  over  the  edge  of  a  ruler  to  raise  the  weight  1  ft. 
How  much  work  ?  Is  the  amount  of  pressure  given  by  the 
spring  balance  or  by  the  pound  weight  ?  Pull  the  weight  along 
the  top  of  the  desk  1  ft.  How  much  work  ?  Hook  the  spring 
balance  under  the  edge  of  the  desk  and  pull  2  lb.  How  much 
work  ?  Drop  the  weight  1  ft.  How  much  work  ought  the 
weight  to  do  when  it  strikes  the  floor  ? 

A  boy  weighing  60  lb.  climbs  up  a  ladder  10  ft.  vertically. 
How  much  work  ?  How  much  work  is  done  when  he  comes 
down  the  ladder  ? 

16 


WORK  AND  POWER  17' 

A  boy  weighing  60  lb.  walks  up  a  flight  of  stairs.  How  much 
work  has  he  done  when  he  has  risen  10  ft.  ?  Why  should  the 
answer  be  the  same  as  in  the  preceding  problem  ? 

A  stone  weighing  50  lb.  is  on  the  roof  of  a  shed  10  ft.  from 
the  ground.  How  much  work  was  done  to  get  it  in  that  posi- 
tion ?  If  it  is  pushed  oif,-how  much  work  ought  it  to  do  when 
it  strikes  the  ground  ?  Why  ought  the  two  answers  to  be  the 
same  number  of  foot  pounds  ? 

15.  Power.  Time  is  not  involved  in  work.  A  man  may  take 
4  hr.  or  10  hr.  to  raise  a  ton  of  coal  15  ft. ;  in  either  case  he 
has  done  30,000  ft.  lb.  of  work.  But  in  the  first  case  he  is 
doing  work  at  the  rate  of  125  ft.  lb.  per  minute,  while  in  the 
second  case  he  is  working  at  the  rate  of  50  ft.  lb.  per  minute. 
To  compare  the  work  of  men  or  machines,  or  to  determine  the 
usefulness  of  a  machine,  it  is  necessary  to  take  into  considera- 
tion the  time  required  for  the  work. 

Power  is  the  rate  of  doing  work.  Thus  if  an  electric  crane 
raises  a  steel  beam,  weighing  500  lb.,  80  ft.  in  2  min.,  its  rate 

of  work  is =  20,000  ft.  lb.  per  minute. 

The  unit  of  power,  the  horse  jiower,  is  the  power  required  to 
do  work  at  the  rate  of  33,000  ft.  lb.  per  minute.  If  a  steam 
crane  lifts  90  T.  of  coal  11  ft.  in  20  min.,  neglecting  friction, 
the  horse  power  of  the  engine  is 

^  2000  X  90  X  11  ^  o 
^'  ^^'         33,000  X  20 

When  we  speak  of  the  horse  power  of  an  engine  we  usually 
mean  the  indicated  horse  power  (i.  h.  p.),  which  is  calculated 
from  the  dimensions  of  the  cylinder  and  the  mean  effective 
steam  pressure  obtained  from  the  indicator  card.  The  horse 
power  actually  available  for  work  is  called  the  brake  horse 
power  (b.  h.  p.),  and  is  determined  by  the  Proiiy  brake  or  a 
similar  device. 


18  APPLIED  MATHEMATICS 

The  horse  power  of  a  steam  engine  is  given  by  the  equation 

,  p  -l-  a-n 

^•P'=    33,000   ' 

where  p  =  mean  effective  pressure  in  pounds  per  square  inch, 
I  =  length  of  stroke  in  feet, 
a  =  area  of  piston  in  square  inches, 
n  =  number  of  strokes  per  minute,  or  twice  the  number 
of  revolutions  per  minute. 

PROBLEMS 

In  these  problems  no  account  is  taken  of  friction  and  other 
losses. 

1.  If  a  man  exerts  a  pressure  of  56  lb.  while  wheeling  a 
barrow  load  of  earth  25  ft.,  find  the  number  of  foot  pounds  of 
work  he  does. 

2.  How  much  work  is  done  by  a  steam  crane  in  lifting  a 
block  of  stone  weighing  1.2  T.  30  ft.  ? 

3.  A  hole  is  punched  through  an  iron  plate  ^  in.  thick.  If 
the  punch  exerts  a  uniform  pressure  of  40  T.,  find  the  work 
done. 

4.  A  horse  hauling  a  wagon  exerts  a  constant  pull  of  75  lb. 
and  travels  at  the  rate  of  4  mi.  per  hour.  How  much  work  will 
the  horse  do  in  3  hr.  ?  If  the  driver  rides  on  the  wagon,  how 
much  work  does  he  do  ? 

5.  A  man  weighing  150  lb.  carries  50  lb.  of  brick  to  the 
top  of  a  building  40  ft.  high.  How  much  work  has  he  done 
(a)  in  getting  himself  to  the  top  ?  (b)  in  carrying  the  brick  ? 
How  much  work  is  done  on  his  return  trip  down  the  ladder  ? 

6.  If  a  pump  is  raising  2000  gal.  of  water  per  hour  from 
the  bottom  of  a  mine  400  ft.  deep,  how  many  foot  pounds  of 
work  are  done  in  2  hr.  ?    (A  gallon  of  water  weighs  8.3  lb.) 

7.  How  many  gallons  of  water  would  be  raised  per  minute 
from  a  mine  600  ft.  deep  by  an  engine  of  180  h.  p.  ? 


AVORK  AND  POWER  19 

8.  The  plunger  of  a  force  pump  is  4  in.  in  diameter,  the 
length  of  the  stroke  is  3  ft.,  and  the  pressure  of  the  water  is 
40  lb.  per  square  inch.    Find  the  work  done  in  one  stroke. 

9.  A  well  6  ft.  in  diameter  is  dug  30  ft.  deep.  If  the  earth 
weighs  125  lb.  per  cubic  foot,  find  the  work  done  in  raising  the 
material. 

10.  A  basement  20  ft.  by  15  ft.  is  filled  with  water  to  a 
depth  of  4  ft.  How  much  work  is  done  in  pumping  the  water 
to  the  street  level,  6  ft.  above  the  basement  floor  ?  (The  aver- 
age distance  which  the  water  is  lifted  is  4  ft.) 

11.  A  chain  40  ft.  long  weighing  101b.  per  foot  is  hanging 
vertically  in  a  shaft.  Construct  a  curve  to  show  the  work  done 
on  each  foot  in  lifting  the  chain  to  the  surface.  (Assume  that 
the  first  foot  is  lifted  J  ft.,  the  second  1^  ft.,  and  so  on.)  What 
is  the  total  work  done  in  lifting  the  chain  ? 

12.  How  much  work  is  done  in  rolling  a  200-lb.  barrel  of 
flour  up  a  plank  to  a  platform  6  ft.  high  ? 

13.  A  boy  who  can  push  with  a  force  of  40  lb.  wants  to  roll 
a  barrel  weighing  120  lb.  into  a  wagon  3  ft.  high.  How  long  a 
plank  must  he  use  ?   (Length  of  plank  x  40  =  3  x  120.  Why  ?) 

14.  A  man  can  just  lift  a  barrel  weighing  200  lb.  into  a  wagon 
3^  ft.  high.  How  much  work  does  he  do  ?  How  long  a  plank 
would  he  need  to  roll  up  a  barrel  weighing  400  lb.  ?    600  lb.  ? 

15.  A  horse  drawing  a  cart  along  a  level  road  at  the  rate  of 
3  mi.  per  hour  performs  42,000  ft.  lb.  of  work  in  5  min.  Find 
the  pull  in  pounds  that  the  horse  exerts  in  drawing  the  cart. 

16.  A  horse  attached  to  a  capstan  bar  12  ft.  long  exerts  a  pull 
of  120  lb.  How  much  work  is  done  in  going  around  the  circle 
100  times  ? 

17.  How  long  will  it  take  a  man  to  pump  800  cu.  ft.  of 
water  from  a  depth  of  16  ft.  if  he  can  do  2000  ft.  lb.  of  work 
per  minute  ? 

18.  How  much  work  can  a  2  h.  p.  electric  motor  do  in 
10  min.  ?  in  15  sec,  ? 


20  APPLIED  MATHEMATICS 

19.  What  is  the  horse  power  of  an  electric  crane  that  lifts 
4  T.  of  coal  30  ft.  per  minute  ?  If  40  per  cent  of  the  power  is 
lost  in  friction  and  other  ways,  what  horse  power  would  be 
required  ? 

20.  Find  the  horse  power  of  an  engine  that  would  punip 
40  cu.  ft.  of  water  per  minute  from  a  depth  of  420  ft.,  if  20 
per  cent  of  the  power  is  lost. 

21.  A  locomotive  exerts  a  pull  of  2  T.  and  draws  a  train  at 
a  speed  of  20  mi.  per  horn*.    Find  the  horse  power. 

22.  The  weight  of  a  train  is  120  T.  and  the  drawbar  pull  is 
7  lb.  per  ton  of  load.  Find  the  horse  power  required  to  keep 
the  train  running  at  the  rate  of  30  mi.  per  hour. 

23.  The  drawbar  pull  of  a  locomotive  pulling  a  passenger 
train  at  a  speed  of  60  mi.  per  hour  is  5500  lb.  At  what  horse 
power  is  the  engine  working  ? 

24.  What  is  the  horse  power  of  Niagara  Falls  if  700,000  T. 
of  water  pass  over  every  minute  and  fall  160  ft.  ? 

25.  If  a  10  h.  p.  pump  delivers  100  gal.  of  water  per  minute, 
to  what  height  can  the  water  be  pmnped  ? 

26.  A  derrick  used  in  the  construction  of  a  building  lifts  an 
I-beam  weighing  2  T.  50  ft.  j)er  minute.  What  is  the  horse 
power  of  the  engine,  if  20  per  cent  of  the  power  is  lost  ? 

27.  In  a  certain  mine  400  T.  of  ore  are  raised  from  a  depth 
of  1000  ft.  during  a  day  shift  of  10  hr.  Neglecting  losses,  what 
horse  power  is  required  to  raise  the  ore  ? 

28.  In  supplying  a  town  with  water  8,000,000  gal.  are  raised 
daily  to  an  average  height  of  120  ft.  What  is  the  horse  power 
of  the  engine  ? 

29.  A  belt  passing  around  two  pulleys  moves  with  a  velocity 
of  15  ft.  per  second.  Find  the  horse  power  transmitted  if  the 
difference  in  tension  of  the  two  sides  of  the  belt  is  1100  lb. 

30.  What  is  the  difference  in  tension  of  the  two  sides  of  a 
belt  that  is  running  3600  ft.  per  minute  and  is  transmitting 
280  h.  p.  ? 


WORK  AND  POWER 


21 


31.  Find  the  number  of  revolutions  per  minute  which  a 
driving  pulley  2  ft.  in  diameter  must  make  to  transmit  12  h.  ])., 
if  the  driving  force  of  the  belt  is  250  lb. 

32.  A  belt  transmits  60  h.  p.  to  a  pulley  20  in.  in  diameter, 
running  at  250  r.  p.  m.  What  is  the  difference  in  pounds  of 
the  tension  on  the  tight  and  slack  sides  ? 

33.  In  a  power  test  of  an  elec- 
tric motor  a  friction  brake  con- 
sisting of  a  strap,  a  weight,  and 
a  spring  balance  was  used.  The 
radius  of  the  pulley  was  2  in., 
the  pull  waa  7  lb.,  and  the  speed 
of  the  shaft  was  1800  r.  p.  m. 
What  horse  power  did  the  test 
give  ? 


Fig.  4.    Friction  Bkakk 


Solution. 


h.  p.  = 


2  X  22  X  2  X  1800  x  7 
7  X  12  X  33,000 


A. 


34.  In  a  power  test  of  a  small  dynamo  the  pull  was  6  lb. 
and  the  speed  was  1500  r.  p.  m.  If  the  radius  of  the  driving 
pulley  was  3  in.,  find  the  horse  power. 

35.  In  testing  a  motor  with  a  Prony  brake  the  pull  was  12  lb., 
length  of  brake  arm  was  18  in.,  and  the  speed  was  500  r.  p.  m. 
Find  the  horse  power.  .  q 


Prony  Brake 


36.  In  testing  a  Corliss  engine  running  at  100  r.  p.  m.  a 
Prony  brake  was  used.    The  lever  arm  was  10.5  ft.  and  the 


22  APPLIED  MATHEMATICS 

pressure  exerted  at  the  end  of  the  arm  was  2000  lb.  What  was 
the  horse  power  ?  In  a  second  test  with  a  pressure  of  2200  lb. 
the  speed  was  90  r.  p.  m.    Find  the  horse  power. 

37.  Calculate  the  horse  power  of  a  steam  engine  from  the 
following  data :  stroke,  2  ft. ;  diameter  of  cylinder,  16  in. ; 
r.  p.  m.,  100 ;  mean  effective  pressure,  60  lb.  per  square  inch. 

38.  The  diameter  of  the  cylinder  of  an  engine  is  20  in.  and 
the  length  of  stroke  is  4  ft.  Find  the  horse  power  if  the  engine 
is  making  60  r.  p.  m.  with  a  mean  effective  pressure  of  601b. 
per  square  inch. 

39.  Find  the  horse  power  of  a  locomotive  engine  if  the  mean 
effective  pressure  is  90  lb.  per  square  inch,  each  of  the  two 
cylinders  is  16  in.  in  diameter  and  24  in.  long,  and  the  driv- 
ing wheels  make  120  r.  p.  m. 

40.  On  a  side-wheel  steamer  the  engine  has  a  6-ft.  stroke, 
the  shaft  makes  35  r.  p.  m.,  the  mean  effective  pressure  is  30  lb. 
per  square  inch,  and  the  diameter  of  the  cylinder  is  4  ft.  Find 
the  horse  power  of  the  engine. 

41.  Find  the  horse  power  of  a  marine  engine,  the  diameter  of 
the  cylinder  being  5  ft.  8  in.,  length  of  stroke  5  ft.,  r.  p.  m.  15, 
and  mean  effective  pressure  30  lb.  per  square  inch. 

42.  The  diameter  of  the  cylinder  of  a  514  h.  p.  marine  engine 
is  5  ft.,  length  of  stroke  6  ft.,  r.  p.  m.  20.  Find  the  mean  effec- 
tive pressure. 

43.  Find  the  diameter  of  the  cylinder  of  a  525  h.  p.  steam 
engine :  stroke,  6  ft. ;  r.  p.  m.,  15 ;  mean  effective  pressure,  25  lb. 
per  square  inch. 

44.  What  diameter  of  cylinder  will  develop  10.3  h.  p.  with  a 
6-in.  stroke,  300  r.  p.  m.,  and  a  mean  effective  pressure  of  90  lb. 
per  square  inch  ? 

45.  The  cylinder  of  a  55  h.  p.  engine  is  12  in.  in  diameter 
and  28  in.  long.  If  the  mean  effective  pressure  is  60  lb.  per 
square  inch,  find  the  number  of  revolutions  per  minute. 


WORK  AND  POWER  23 

16.  Mechanical  efficiency  of  machines.  The  useful  work 
given  out  by  a  machine  is  always  less  than  the  work  put  into 
it  because  of  the  losses  due  to  the  weight  of  its  parts,  friction, 
and  so  on.  If  there  were  no  losses,  the  efficiency  would  be  100 
per  cent. 

The  efficiency  of  a  machine  is  the  quotient  obtained  by  divid- 
ing the  useful  work  of  the  machine  by  the  work  put  into  it. 

■n/v,  .  Output 

^^'"""^y^^^t- 

_,^  .  -  .  Brake  horse  power 

Mijjiciency  of  a  steam  engine  =  - — —-^ — 

Indicated  horse  jJOiver 

In  general  the  efficiency  of  a  machine  increases  with  the  load 
up  to  a  certain  point,  and  then  falls  off.  Small  engines  are 
often  run  at  an  efficiency  of  less  than  80  per  cent ;  large 
engines  usually  have  an  efficiency  of  85  to  90  per  cent. 

PROBLEMS 

1.  A  steam  crane  working  at  3  h.  p.  raises  a  block  of  granite 
weighing  8  T.,  50  ft.  in  12  min.   Find  the  efficiency  of  the  crane. 

„  ^  ^     ^      2000  X  8  X  50  ,^  ,, 

Solution.   Output  = — ft.  lb.  per  minute. 

Input  =  3  X  33,000  ft.  lb.  per  minute. 

,,„  .  2000  X  8  X  50 

Efficiency  = =  67  per  cent. 

^      12  X  3  X  33,000  ^ 

2.  A  6  h.  p.  electric  crane  lifts  a  machine  weighing  15  T. 
at  the  rate  of  5  ft.  per  minute.    What  is  the  efficiency  ? 

3.  An  engine  of  150  h.  p.  is  raising  1000  gal.  of  water  per 
minute  from  a  mine  500  ft.  deep.  Find  the  efficiency  of  the 
pumping  system. 

4.  An  elevator  motor  of  50  h.  p.  raises  the  car  aijd  its  load, 
2800  lb.  in  all,  120  ft.  in  15  sec.    Find  the  efficiency. 

5.  How  long  will  it  take  a  20  h.  p.  engine  to  raise  2  T.  of 
coal  from  a  mine  300  ft.  deep,  if  the  efficiency  is  80  per  cent  ? 


24  APPLIED  MATHEMATICS 

6.  What  is  the  efBciency  of  an  engine  if  the  indicated  horse 
power  is  250  and  the  brake  horse  power  is  225  ? 

7.  In  lifting  a  weight  of  256  lb.  20  ft.  by  means  of  a  tackle 
a  man  hauls  in  64  ft.  of  rope  with  an  average  pull  of  110  lb. 
Find  the  efficiency  of  the  tackle. 

8.  The  efficiency  of  a  set  of  pulleys  is  75  per  cent.  How 
many  pounds  must  be  the  pull,  acting  through  88  ft.,  to  raise 
a  load  of  525  lb.  a  distance  of  20  ft.  ? 

9.  A  pump  of  10  h.  p.  raises  54  cu.  ft.  of  water  per  minute 
to  a  height  of  80  ft.   What  is  its  efficiency  ? 

10.  A  steam  crane  unloads  coal  from  a  vessel  at  the  rate  of 
20  T.  in  8  min.,  and  lifts  it  a  total  distance  of  24  ft.  If  the 
combined  efficiency  of  the  engine  and  crane  is  70  per  cent, 
what  is  the  horse  power  of  the  engine  ? 

11.  Find  the  power  required  to  raise  4800  gal.  of  water  60  ft. 
in  2  hr.  if  the  efficiency  of  the  pump  is  60  per  cent. 

12.  A  centrifugal  pump  whose  efficiency  when  lifting  water 
12  ft.  is  62  per  cent,  is  required  to  lift  18  cu.  ft.  per  second  to 
a  height  of  12  ft.    What  must  be  its  horse  power  ? 

13.  A  dock  200  ft.  long  and  50  ft.  wide  is  filled  with  water 
to  a  depth  of  30  ft.  It  is  emptied  in  40  min.  by  a  centrifugal 
pump  which  delivers  the  water  40  ft.  above  the  bottom  of  the 
dock.  If  the  combined  efficiency  of  the  engine  and  pump  is 
70  per  cent,  what  is  the  horse  power  of  the  engine  ?  (A  cubic 
foot  of  sea  water  weighs  64  lb.  The  average  distance  which 
the  water  is  lifted  is  25  ft.) 

14.  A  steam  engine  having  a  cylinder  10  in.  in  diameter  and 
a  stroke  of  24  in.  makes  80  r.  p.  m.  and  gives  a  brake  horse 
power  of  34  h.  p.  If  the  mean  effective  pressure  is  50  lb.  per 
square  inch,  find  the  efficiency. 

15.  In  testing  a  Corliss  engine  running  at  80  r.  p.  m.  a  Prony 
brake  was  used.  The  lever  arm  was  10.5  ft.  and  the  pressure 
at  the  end  of  the  arm  was  1600  lb.  The  indicated  horse  power 
was  290.    Find  the  efficiency  of  the  engine. 


WORK  AND  POWER 


25 


16.  The  efficiency  of  a  boiler  is  70  per  cent  and  of  the  en- 
gine 80  per  cent.    What  is  the  combined  efficiency  ? 

Solution.  .80  x  .70  =  56  per  cent. 

17.  Power  is  obtained  from  a  motor.  If  the  efficiency  of  the 
motor  is  88  per  cent,  of  the  dynamo  85  per  cent,  and  of  the 
engine  86  per  cent,  what  is  the  combined  efficiency  ? 

18.  The  engine  which  furnishes  power  for  a  centrifugal 
pump  has  an  indicated  horse  power  of  14  and  an  efficiency  of 
88  per  cent.  What  is  the  efficiency  of  the  pump  if  it  is  raising 
3000  gal.  of  clear  water  12  ft.  high  per  minute  ? 

19.  In  a  test  to  find  the  efficiency  of  a  set  of  pulleys  the  fol- 
lowing results  were  obtained.    Construct  the  efficiency  curve. 


Weight  lif  ted(pounds) 

5 

10 

15 

20 

25 

30 

35 

Distance  (feet)      .     . 

1 

1 

1 

1 

1 

1 

1 

Pull  in  pounds      .     . 

3 

5 

6.5 

8 

9.5 

11 

12.8 

Distance  (feet)      .     . 

3 

3 

3 

3 

3 

3 

3 

20.  In  a  test  to  determine  the  relative  efficiency  of  centrif- 
ugal and  reciprocating  pumps  the  following  results  were  ob- 
tained.   Construct  the  efficiency  curves. 


Lift  in  feet 

Efficiency  of  reciprocat- 
ing pump  (per  cent)    . 

Efficiency  of  centrifugal 
pump  (per  cent) .    .    . 


50 


56 


280 

85 


21.  In  a  laboratory  experiment  to  determine  the  efficiency  of 
a  set  of  pulleys  the  following  results  were  obtained.  Construct 
the  efficiency  curve. 


Load  in  grains 
Efficiency 


40 
13.2 


90 
26.5 


140 
36.0 


190 
43.2 


240 
49.0 


290 
53.6 


340 
57.4 


390 
60.6 


440 

63.1 


490 
65.5 


26 


APPLIED  MATHEMATICS 


22.  The  following  results  were  obtained  in  an  experiment  to 
find  the  efficiency  of  a  set  of  differential  chain  pulley  blocks. 
Find  the  efficiency  in  each  test  and  construct  the  efficiency 


curve. 


Load  in  pounds     . 

7 

21 

35 

49 

70 

98 

112 

126 

140 

Distance  (feet)  .     . 

1 

1 

1 

1 

1 

1 

1 

1 

1 

Pull  in  pounds 

3.22 

5.73 

8.40 

11.03 

15.13 

20.17 

23.17 

26.00 

29.05 

Distance  (feet)  .     . 

16 

16 

16 

16 

16 

16 

16 

16 

16 

23.  Find  the  efficiency  of  the  following  engines  : 

No. 

Type 

Pressure 

in  lb. 
per  sq.  in. 

Stroke  in 
inches 

Diameter  of 

cylinder  in 

inciies 

Revolutions 
per  minute 

Brake 
horse 
power 

1 

Marine .     .     . 

37 

168 

110 

17 

4440 

2 

Marine .     .     . 

25 

72 

70 

15 

441 

3 

Corliss  .     .     . 

90 

48 

30 

85 

1180 

4 

Gas  engine  *  . 

62 

16 

12 

150 

18 

5 

Locomotive    . 

80 

24 

17 

260 

504 

6 

High-speed     . 

50 

16 

12 

246 

100 

7 

Medium-speed 

75 

36 

24 

100 

533 

*  Explosion  every  two  revolutions. 


CHAPTER  IV 


LEVERS  AND  BEAMS 


17.  Law  of  the  lever.  A  rigid  rod  movable  about  a  fixed 
point  may  be  held  in  equilibrium  by  two  or  more  forces.  To 
find  the  relation  between  these  forces  when  the  lever  is  in 
a  state  of  balance,  we  will  make  a  few  experiments. 


I 


Tr' 


EXERCISES 

1.  Balance  a  meter  stick  at  its  center ;  suspend  on  it  two 
unequal  weights  so  that  they  balance.  Which  weight  is  nearer 
the  center?  Multiply  each 
weight  by  its  distance  from 
the  center  and  compare  the  c 
products.  Do  this  with 
several  pairs  of  weights. 
What  seems  to  be  true  ? 

2.  Balance  a  meter  stick 
as  before,  and  put  a  500-g. 
weight  12  cm.  from  the  cen- 
ter ;  then  in  turn  put  on  the 
following  weights  so        Law  of  the  lever :  p 

that    each   balances    p         

the    500-g.    weight.   / 
In  each  case  record 


,^r 


Fig.  6 


PF  =  IV 


WF. 


~K 


Fig.  7 


the  distance  from  the  weight  to  the  center. 


Grams  .... 
Measured  distance  . 
Computed  distance , 


450 


400 


350 


300 


250 


200 


150 


120 


27 


28  APPLIED  MATHEMATICS 

Locate  a  point  on  squared  paper  for  each  weight.  Units : 
horizontal,  1  large  square  =  5  cm. ;  vertical,  1  large  square  = 
50  g.  Draw  a  smooth  curve  through  the  points.  Take  some 
intermediate  points  on  the  curve  and  test  the  readings  by  put- 
ting the  weights  on  the  meter  stick.  On  the  same  sheet  of 
squared  paper  draw  the  curve,  using  the  computed  distances. 

3.  Suspend  two  unequal  weights  on  one  side  of  the  center 
and  balance  them  with  one  weight.  What  is  the  law  of  the 
lever  for  this  case  ? 

Definitions.  The  point  of  support  is  called  the  fulcrum.  The 
product  of  a  weight  and  its  distance  from  the  fulcrum  is  called 
the  leverage  of  the  weight.  The  quotient  of  the  length  of  one 
arm  of  the  lever  divided  by  the  length  of  the  other  is  called 
the  mechanical  advantage  of  the  lever. 

The  force  which  causes  a  lever  to  turn  about  the  fulcrum 
may  be  called  the  power  (p),  and  the  body  which  is  moved  may 
be  called  the  weight  (w). 

18.  Three  classes  of  levers.  Levers  are  divided  into  three 
classes,  according  to  the  position  of  the  power,  fulcrum,  and 

^^^S^^-      ^  P r w 

i-  7C 


^  p  w  r 

^ zs: 

jjj:Yi E S. 

Fig.  8 

Class  I.  When  the  fulcrum  is  between  the  power  and  the 
weight.    Name  some  levers  of  this  class. 

Class  II.  When  the  weight  is  between  the  fulcrum  and  the 
power.   Name  some  levers  of  this  class. 

Class  III.  When  the  power  is  between  the  fulcrum  and 
the  weight.   Name  some  levers  of  this  class. 


LEVERS  AND  BEAMS  29 

19.  Levers  of  the  first  class.  With  a  lever  of  this  class  a 
large  weight  may  be  lifted  by  a  small  power;  time  is  lost 
while  mechanical  advantage  is  gained. 

PROBLEMS 

In  these  problems  on  levers  of  the  first  class  either  the  lever 
is  "  weightless,"  —  that  is,  it  is  supposed  to  balance  at  the  ful- 
crum, —  or  else  the  weight  of  the  lever  is  neglected.  Draw  a 
diagram  for  each  problem. 

1.  What  weight  12  in.  from  the  fulcrum  will  balance  a  6-lb. 
weight  14  in.  from  the  fulcrum  ? 

Solution.   Let  tv  =  the  weight. 

12w  =  6xl4. 
10  =  7. 
Check.  12  X  7  =  6  X  14. 

84  =  84. 

2.  How  far  from  the  fulcrum  must  a  7-lb.  weight  be  placed 
to  balance  a  4-lb.  weight  35  cm.  from  the  fulcriun  ? 

3.  What  is  the  weight  of  an  object  10  in.  from  the  fulcrum, 
if  it  balances  a  weight  of  3  lb.  14.4  in.  from  the  fulcrum  ? 

4.  A  meter  stick  is  balanced  at  the  center.  On  one  side  are 
two  weights  of  10  lb.  and  4  lb.,  4  in.  and  1\  in.  from  the  ful- 
crum respectively.  How  far  from  the  fulcrum  must  a  7-lb. 
weight  be  placed  to  balance  ? 

5.  Two  books  weighing  250  g.  and  625  g.  are  suspended 
from  a  meter  stick  to  balance.  The  heavier  book  is  12  cm.  from 
the  center.    How  far  is  the  other  book  from  the  center  ? 

6.  A  5-g.  and  a  50-g.  weight  are  placed  to  balance  on  a 
meter  stick  suspended  at  its  center.  If  the  leverage  is  100, 
how  far  is  each  weight  from  the  center? 

7.  An  iron  casting  weighing  6  lb.  is  broken  into  two  pieces 
which  balance  on  a  meter  stick  when  the  mechanical  advantage 
is  4.    Find  the  weight  of  each  piece. 


30 


APPLIED  MATHEMATICS 


8.  Two  boys  weighing  96  and  125  lb.  play  at  teeter.  If  the 
smaller  boy  is  8  ft.  from  the  fulcrum,  how  far  is  the  other  boy 
from  that  point  ? 

9.  Two  boys  playing  at  teeter  weigh  67  lb.  and  120  lb.  and 
are  7  ft.  and  6  ft.  respectively  from  the  fulcrum.  Where  must 
a  boy  weighing  63  lb.  sit  to  balance  them  ? 

10.  Two  bolts  weighing  together  392  g.  balance  when  j^laced 
50  cm.  and  30  cm.  respectively  from  the  fulcrum.  Find  the 
weight  of  each. 

11.  A  boy  weighing  95  lb.  has  a  crowbar  6  ft.  long.  How 
can  he  arrange  things  to  raise  a  block  of  granite  weighing 
280  lb.  ? 

12.  A  lever  15  ft.  long  balances  when  weights  of  72  lb.  and 
108  lb.  are  hung  at  its  ends.    Find  the  position  of  the  fulcrum. 


Problems  in  which  xhe  Weight  of  the  Lever 
is  included 

Exercise  1.  Test  a  meter  stick  to  see  if  it  balances  at  the 
center.    If  it  does  not,  add  a  small  weight  to  make  it  balance. 

Weigh  the  meter  stick.    It  is  found  to  weigh  162  g. 
T^f  ~  -^'^^  S-  P^'"  centimeter. 
Attach  a  200-g.  weight  to  one 
end  and  balance  as  in  Fig.  9.  ' 

The  length  of  FW  =  22.4  cm. 

The  length  of   PF  =  77.6  cm. 

The  weight  of 
FW =22 A  xl.Q2=   36.2  g. 

The  weight  of 
PF  =  77.6  X  1.62  =  125.7  g. 


J 


D 


161.9  g.  Check. 


Fig.  9 


The  200-g.  weight  and  the  short  length  of  the  meter  stick  balance 
the  long  part.  Let  us  suppose  that  the  weight  of  each  part  is  con- 
centrated at  the  center  of  the  part,  and  apply  the  law  of  the  lever. 


LEVERS  AND  BEAMS  81 

77  fi  22  4 

125.7  X  1^  =  36.2  X  ^  +  200  x  22.4. 
2  2 

125.7  X  38.8=  36.2  x  11.2  +  4480. 

4877  =  4885. 

This  checks  as  near  as  can  be  expected  in  experimental  work. 
The  measurements  were  made  to  three  figures  and  the  results  differ 
only  by  one  in  the  third  place. 

Hence  when  a  uniform  bar  is  used  as  a'  lever  we  may  assume 
that  the  weight  of  each  part  is  concentrated  at  the  mid-point  of 
the  part. 

A  shorter  method  of  solution  is  to  consider  the  weight  of  the 
lever  as  concentrated  at  its  center.    Thus,  in  the  preceding  exercise  : 

200  x  i^lF=  162  (50  -  FW). 
Solving,  FW  =  22.4. 

Exercise  2.  Make  a  similar  test  with  a  metal  bar. 


PROBLEMS 

1.  One  end  of  a  stick  of  timber  weighing  10  lb.  per  linear 
foot,  and  14  ft.  long,  is  placed  under  a  loaded  wagon.  If  the 
fulcrum  is  2  ft.  from  the  end,  how  many  pounds  does  the  tim- 
ber lift  when  it  is  horizontal  ? 

Solution.    Let  x  —  number  of  pounds  lifted. 

2  X  +  20  X  1  =:  120  X  6. 

x  =  3501b.  •    ■ 

Check.  2  X  350  +  20  =  120  x  6. 

720  =  720. 

2.  A  lever  20  ft.  long  and  weighing  12  lb.  per  linear  foot  is 
used  to  lift  a  block  of  granite.  The  fulcrum  is  4  ft.  from  one 
end  and  a  man  weighing  180  lb.  puts  his  weight  on  the  other 
end.    How  many  pounds  are  lifted  on  the  stone  ? 

3.  A  uniform  lever  12  ft.  long  and  weighing  36  lb.  balances 
upon  a  fulcrum  4  ft.  from  one  end  when  a  load  of  x  lb.  is 
hung  from  that  end.    Find  the  value  of  x. 


32 


APPLIED  MATHEMATICS 


4.  A  uniform  lever  10  ft.  long  balances  about  a  point  1  ft. 
from  one  end  when  loaded  at  that  end  with  50  lb.  What  is  the 
weight  of  the  lever  ? 


Fig.  10 


Solution. 

Let 

X 

=  weight  of  a 

linear  f 

50  X 

1  +  X 

x| 

X 

10  X 

=  9x  X  41. 
=  lilb. 
=  121  lb. 

Check. 

50  X 

1  +  ^ 

xf 
50| 

=  9  X  f  X  41. 
=  50f . 

Second 

So 

LUTION. 

Let 

X 

=  weight  of  the  lever. 

X 

X  4 
4x 

X 

=  50  X  1. 
=  50. 
=  121  lb. 

Check. 

12A 

X  4 

=  50. 

5.  A  man  weighing  180  lb.  stands  on  one  end  of  a  steel 
rail  30  ft.  long  and  finds  that  it  balances  over  a  fulcrum  at 
a  point  2  ft.  from  its  center.  What  is  the  weight  of  the  rail 
per  yard  ? 

6.  A  teeter  board  16  ft.  long  and  weighing  32  lb.  balances  at 
a  point  7  ft.  from  one  end  when  a  boy  weighing  80  lb.  is  seated 
1  ft.  from  this  end  and  a  second  boy  1  ft.  from  the  other  end. 
How  much  does  the  second  boy  weigh  ? 

7.  A  uniform  lever  12  ft.  long  balances  at  a  point  4  ft.  from 
one  end  when  30  lb.  are  hung  from  this  end  and  an  unknown 
weight  from  the  other.  If  the  lever  weighs  24  lb.,  find  the 
unknown  weight. 

20.  Levers  of  the  second  class.  With  a  lever  of  the  first 
class  the  weight  moves  in  a  direction  opposite  to  that  in  which 
the  power  is  applied.    How  is  it  with  a  lever  of  the  second  class  ? 


LEVERS  AND  BEAMS 


33 


EXERCISES 

1.  Place  a  meter  stick  as  shown  in  Fig.  11,  and  put  weights 
in  the  other  pan  to  balance.  This  arrangement  makes  a  weight- 
less lever. 

(a)  Put  100  g.  18  in.  from  F.  How  many  grams  are  required 
to  balance  it  ? 


yy 


Fig.  11 


Solution. 


p-PF=io-  WF. 
36/;  =  100  X  18. 
p  =  bOg. 

Check  by  putting  a  50-g.  weight  in  the  other  pan. 

{h)  Put  100  g.  9  in.  from  F  and  find  p.  Check, 
(c)  Put  100  g.  27  in.  from  F  and  find  p.  Check. 
{d)  Put  200  g.  12  in.  from  F  and  find  p.    Check. 

2.  Lay  a  uniform  metal  bar  2  or  3  ft.  long  on  the  desk  and 
lift  one  end  with  a  spring  balance.  Compare  the  reading  with 
the  weight  of  the  bar.  Make  two  or  three  similar  tests.  What 
seems  to  be  true  ?  Where  is  the  fulcrum  ?  Where  is  the 
power?  Where  is  the  weight  with  reference  to  the  fulcrum 
and  power  ?  Suppose  the  weight  of  the  bar  to  be  concentrated 
at  the  center  and  see  if  the  law  of  the  lever  p  ■  PF  =  w  •  WF 
holds  true. 

3.  Place  a  2-1  b.  weight  on  a  meter  stick  lying  on  the  desk 
at  distances  of  (a)  40  cm.,  (b)  50  cm.,  (c)  60  cm.,  and  {(I)  80  cm. 
from  onei  end.  In  each  case  compute  the  pull  required  to  lift 
the  other  end  of  the  meter  stick.  Check  by  lifting  with  a  spring 
balance. 

4.  Construct  a  graph  to  show  the  results  obtained  in  Exer- 
cise 3.   Why  should  it  be  a  straight  line  ? 


34  APPLIED  MATHEMATICS 

PROBLEMS 

1.  A  lever  6  ft.  long  has  the  fulcrum  at  one  end.  A  weight 
of  120  lb.  is  placed  on  the  lever  2  ft.  from  the  fulcrum.  How- 
many  pounds  pressure  are  required  at  the  other  end  to  keep  the 
lever  horizontal,  (a)  neglecting  the  weight  of  the  lever  ?  (b)  if 
the  lever  is  uniform  and  weighs  20  lb.  ? 

2.  A  man  uses  an  8-ft.  crowbar  to  lift  a  stone  weighing 
800  lb.  If  he  thrusts  the  lever  1  ft.  under  the  stone,  with  what 
force  must  he  lift  to  raise  the  stone  ? 

3.  A  man  is  using  a  lever  with  a  mechanical  advantage  of  6. 
If  the  load  is  1^  ft.  from  the  fulcrum,  how  long  is  the  lever  ? 

4.  A  boy  is  wheeling  a  loaded  wheelbarrow.  The  center  of  the 
total  weight  of  100  lb.  is  2  ft.  from  the  axle  and  the  boy's  hands 
are  5  ft.  from  the  axle.    What  lifting  force  does  he  exert  ? 

5.  A  uniform  yellow-pine  beam  10  ftr  long  weighs  38  lb.  per 
linear  foot.  When  it  is  lying  horizontal  a  man  picks  up  one 
end  of  the  beam.    How  many  pounds  does  he  lift  ? 

6.  To  lift  a  machine  weighing  3000  lb.  a  man  has  a  jack- 
screw  which  will  lift  800  lb.  and  a  beam  12  ft.  long.  If  the 
jackscrew  is  placed  at  one  end  of  the  beam  and  the  other  end 
is  made  the  fulcrum,  how  far  from  the  fulcrum  must  he  attach 
the  machine  in  order  to  lift  it  ? 

21.  Levers  of  the  third  class.  In  all  levers  of  this  class 
the  power  acts  at  a  mechanical  disadvantage  since  it  must  be 
greater  than  the  weight.  Therefore  this  form  of  the  lever  is 
used  when  it  is  desired  to  gain  speed  rather  than  mechanical 
advantage. 

EXERCISES 

Attach  a  nieter  stick  to  the  base  of  the  balance,  as  shown  in 
Fig.  12,  and  let  the  meter  stick  rest  on  a  triangular  block  placed 
in  one  pan  of  the  balance.  Put  weights  in  the  other  pan  to 
balance.    This  makes  a  weightless  lever. 

Let  PF=9  cm. 


LEVERS  AND  BEAMS 


35 


1.  Put  100  g.  18  cm.  from  F.  How  many  grams  are  required 
to  balance  it  ? 

Solution. 

pPF  =  w  WF. 
Qp  =  100  X  18. 
p  =  200. 

Check  by  placing  200  g. 
in  the  pan.  Fig.  12 

2.  Put  100  g.  27  cm.  from  F  and  find  p.    Check. 

3.  Put  50  g.  36  cm.  from  F  and  find  p.      Check. 

4.  Put  50  g.  45  cm.  from  F  and  find  p.     Check. 

5.  Put  one  end  of  a  meter  stick  just  under  the  edge  of  the 
desk.  Hold  the  stick  horizontal  with  a  spring  balance.  Where 
are  the  fulcrum,  weight,  and  power  ?  Where  may  we  consider 
the  weight  of  each  part  of  the  meter  stick  to  be  concentrated  ? 
Weigh  the  meter  stick  and  compute  the  pull  required  to  hold 
it  horizontal.    Check  by  reading  the  spring  balance. 

6.  Make  the  same  experiment  with  a  uniform  metal  bar. 


PROBLEMS 

1.  A  lever  12  ft.  long  has  the  fulcrum  at  one  end.  A  pull  of 
80  lb.  3  ft.  from  the  fulcrum  will  lift  how  many  pounds  at  the 
other  end  ?    Neglect  weight  of  lever. 

2.  The  arms  of  a  lever  of  the  third  class  are  2  ft.  and  6  ft. 
respectively.    How  many  pounds  will  a  pull  of  60  lb.  lift  ? 

3.  With  a  lever  of  the  third  class  a  pull  of  65  lb.  applied 
6  in.  from  the  fulcrum  lifts  a  weight  of  5  lb.  at  the  other  end 
of  the  lever.    How  long  is  the  lever  ?   Neglect  its  weight. 

4.  If  the  mechanical  advantage  of  a  lever  "is  J,  a  pull  of 
how  many  pounds  will  be  required  to  lift  40  lb.  ? 

5.  Construct  a  curve  to  show  the  mechanical  advantage  of  a 
lever  12  ft.  long,  as  the  power  is  applied  1  ft.,  2  ft.,  3i  ft.  .  .  . 
from  the  weight,  the  whole  length  of  the  lever  being  used. 


36 


APPLIED  MATHEMATICS 


22.  Beams.  The  following  exercises  will  show  that  a  straight 
beam  resting  in  a  horizontal  position  on  supports  at  its  ends 
may  be  considered  a  lever  of  the  second  class. 


EXERCISES 

1.  Test  two  spring  balances  to  see  if  they  are  correct.  Weigh 
a  meter  stick.  Suspend  it  on  two  spring  balances,  as  shown  in 
Fig.  13.  Read  each  balance.  Note  that  each  should  indicate 
one  half  the  weight  of  the  meter  stick.  Place  a  200-g.  weight 
at  the  center.   Read  each  balance. 


1 


Fig.  13 

2.  With  the  meter  stick  as  in  Exercise  1,  place  a  200-g. 
weight  10,  20,  30,  ...  90  cm.  from  one  end,  and  record  the 
reading  of  each  balance  after  the  meter  stick  has  been  made 
horizontal.  Construct  a  curve  for  the  readings  of  each  balance 
on  the  same  sheet  of  squared  paper. 

— t- 


fcz 


40 


60- 


FiG.  14 


To  compute  the  reading  of  the  balance  we  need  only  think  of  the 
beam  as  a  lever  of  the  second  class. 

Thus,  when  the  weight  is  40  cm.  from  one  end, 
j9  X  100  =  200  X  60, 
;?  =  120 ; 
and  5  X  100  =  200  x  40, 

<?  =  80. 
Check.  80  +  120  =  200. 


LEVERS  AND  BEAMS 


37 


3.  Suspend  a  500- g.  weight  20  cm.  from  one  end  of  the  meter 
stick.  B-ead  the  balances  after  the  stick  has  been  made  horizontal. 
Correct  for  its  weight.    Compare  with  the  computed  readings. 

4.  Make  similar  experiments  with  metal  bars  and  with  two 
or  three  weights  placed  on  the  bar  at  the  same  time. 


PROBLEMS 


1.  A  man  and  a  boy  are  carrying  a  box  weighing  120  lb.  on 
a  stick  8  ft.  long.  If  the  box  is  3  ft.  from  the  man,  what  weight 
is  each  carrying  ? 


+ 


Solution. 

Arithmetic. 


Algebra.   Let 


Solving, 


W 


A  general  solution. 


Fig.  15 

3  +  5  =  8. 
I  of  120  =    45  lb.,  weight  boy  carries. 
■|  of  120  =    75  lb.,  weight  man  carries. 
120.    Check. 

X  =  number  of  pounds  man  carries. 
y  =  number  of  pounds  boy  carries. 


x-\-  y  =  120. 
3  X  =  5  ^. 
x  =  75. 
2^  =  45. 


-+ 


m 


Fig.  16 

X  (/«  +  n)  =  n  •  W. 

n.W 

X  = 

111  +  n 

y(m-\-  n)  —  m  ■  W. 

m-W 

y  — ■• 

7ft  +  « 


38  APPLIED  MATHEMATICS 

2.  Two  men,  A  and  B,  cany  a  load  of  400  lb.  on  a  pole  be- 
tween them.  The  men  are  15  ft. -apart  and  the  load  is  7  ft, 
from  A.    How  many  pounds  does  each  man  carry  ? 

3.  A  man  and  a  boy  are  to  carry  300  lb.  on  a  pole  9  ft.  long. 
How  far  from  the  boy  must  the  load  be  placed  so  that  he  shall 
carry  100  lb.  ? 

4.  A  beam  20  ft.  long  and  weighing  18  lb.  per  linear  foot 
rests  on  a  support  at  each  end.  A  load  of  1  T.  is  placed  6  ft. 
from  one  end.    Find  the  load  on  each  support. 

•  5.  A  locomotive  weighing  56  T.  stands  on  a  bridge  with  its 
center  of  gravity  30  ft.  from  one  end.  The  bridge  is  80  ft.  long 
and  weighs  100  T. ;  it  is  supported  by  stone  abutments  at  the 
ends.    Find  the  total  weight  supported  by  each  abutment. 

6.  A  man  weighing  192  lb.  walks  on  a  plank  which  rests  on 
two  posts  16  ft.  apart.  Construct  curves  to  show  the  pressure 
on  each  of  the  posts  as  he  walks  from  one  to  the  other. 

MISCELLANEOUS  PROBLEMS 

1.  One  end  of  a  crowbar  6  ft.  long  is  put  under  a  rock,  and 
a  block  of  wood  is  put  under  the  bar  4  in.  from  the  rock.  A 
man  weighing  200  lb.  puts  his  weight  on  the  other  end.  How 
many  pounds  does  he  lift  on  the  rock,  and  what  is  the  pressure 
on  the  block  of  wood  ? 

2.  A  nutcracker  6  in.  long  has  a  nut  in  it  1  in.  from  the 
hinge.  The  hand  exerts  a  pressure  of  4  lb.  What  is  the  pres- 
sure on  the  nut  ? 

3.  What  pressure  does  a  nut  in  a  nutcracker  withstand  if 
it  is  2.8  cm.  from  the  hinge,  and  the  hand  exerts  a  pressure  of 
1.5  kg.  12  cm.  from  the  hinge  ? 

4.  Two  weights,  P  and  Q,  hang  at  the  ends  of  a  weightless 
lever  80  cm.  long.  P  =  1.2  k^.  and  Q  =  3  kg.  Where  is  the 
fulcrum  if  the  weights  balance  ? 


LEVERS  AND  BEAMS 


39 


Fig.  17 


5.  A  man  uses  a  crowbar  7  ft.  long  to  lift  a  stone  weighing 
600  lb.  If  he  thrusts  the  bar  1  ft.  under  the  stone,  with  what 
force  must  he  lift  on  the  other  end  of  the  bar  ? 

6.  A  safety  valve  is  2^  in.  in  diameter  and  the  lever  is 
18  in.  long.  The  distance  from  the  fulcrum  to  the  center  of 
the  valve  is  3  in.  What  weight  must 
be  hung  at  the  end  of  the  lever  so 
that  steam  may  blow  off  at  100  lb. 
per  squai-e  inch,  neglecting  weight 
of  valve  and  lever  ? 

7.  What  must  be  the  length  of 
the  lever  of  a  safety  valve  whose 
area  is  10  sq.  in.,  if  the  weight  is  180  lb.,  steam  pressure  120  lb. 
per  square  inch,  and  the  distance  from  the  center  of  the  valve 
to  the  fulcrum  is  3^  in.  ? 

8.  Find  the  length  of  lever  required  for  a  safety  valve  3  in. 
in  diameter  to  blow  off  at  60  lb.  per  square  inch,  if  the  weight 
at  the  end  of  the  lever  is  75  lb.  and  the  distance  from  the 
center  of  the  valve  to  the  fulcrum  is  2 J  in. 

9.  In  a  safety  valve  of  ^3^  in.  diameter  the  length  of  the 
lever  from  fulcrum  to  end  is  24  in.,  the  weight  is  100  lb.,  and 
the  distance  from  fulcrum  to  center  of  valve  is  3  in.  Find 
the  lowest  steam  pressure  that  will  open  the  valve. 

10.  A  bar  4  m.  long  is  used  by  two  men  to  carry  160  kg.  If 
the  load  is  1.2  m.  from  one  man,  what  weight  does  each  carry  ? 

11.  A  bar  12  ft.  long  and  weighing  40  lb.  is  used  by  two 
men  to  carry  240  lb.  How  many  pounds  does  each  man  carry 
if  the  load  is  5  ft.  from  one  man  ? 

12.  A  man  and  a  boy  have  to  carry  a  load  slung  on  a  light 
pole  12  ft.  long.  If  their  carrying  powers  are  in  the  ratio  8 : 5, 
where  should  the  load  be  placed  on  the  pole  ? 

13.  A  wooden  beam  15  ft.  long  and  weighing  400  lb.  carries 
a  load  of  2  T.  5  ft.  from  one  end.  Find  the  pressure  on  the 
support  at  each  end  of  the  beam. 


40 


APPLIED  MATHEMATICS 


14.  A  beam  carrying  a  load  of  5  T.  3  ft.  from  one  end  rests 
with  its  ends  upon  two  supports  20  ft.  apart.  If  the  beam 
is  uniform  and  weighs  2  T.,  calculate  the  pressure  on  each 
support. 

15.  The  horizontal  roadway  of  a  bridge  is  30  ft.  long  and  its : 
weight  is  6  T.    What  pressure  is  borne  by  each  support  at  the 
ends  when  a  wagon  weighing  2  T.  is  one  third  the  way  across  ? 

16.  An  iron  girder  20  ft.  long  and  weighing  60  lb.  per  foot 
carries  a  distributed  load  of  1800  lb.,  and  two  concentrated 
loads  of  1500.1b.  each  6  ft.  and  12  ft.  respectively  from  one 
support.    Calculate  the  pressure  on  each  support. 

17.  One  end  of  a  beam  8  ft.  long  is  set  solidly  in  the  wall, 
as  in  Fig.  18.    If  the  beam  weighs  40  lb. 

per  linear  foot,  what  is  the  bending  moment 
at  the  wall  ? 

Solution.  The  bending  moment  at  any  [ 
point  A  is  equal  to  the  weight  multiplied  by 
its  distance  from  A .  We  may  assume  that  the 
weight  of  the  beam  is  concentrated  at  its  center 
4  ft.  from  the  wall.  Hence  the  bending  moment 
=  320  X  4  =  1280  lb.  ft. 


Fig.  18 


18.  In  Fig,  18  a  weight  of  800  lb.  is  placed  at  the  end  of 
the  beam  away  from  the  wall.  What  will  be  the  total  bending 
moment  ? 

19.  A  steel  beam  weighing  100  lb.  per  linear  foot  projects 
20  ft.  from  a  solid  wall.  What  is  the  bending  moment  at  the 
wall  ?  What  weight  must  be  placed  at  the  outer  end  to  make 
the  bending  moment  five  times  as  great  ? 

20.  A  stiff  pole  15  ft.  long  sticks  out  horizontally  from  a 
vertical  wall.  It  would  break  if  a  weight  of  30  lb.  were  hung 
at  the  end.  How  far  out  on  the  pole  may  a  boy  weighing  80  lb. 
go  with  safety  ? 

21.  A  steel  beam  15  ft.  long  projects  horizontally  from  a 
vertical  wall.    At  the  end  is  a  weight  of  400  lb.    Construct  a 


LEVERS  AND  BEAMS  41 

curve  to  show  the  bending  moments  of  this  weight  at  various 
points  on  the  beam  from  the  wall  to  the  outer  end. 

Suggestion.  The  bending  moment  at  the  wall  is  400  x  15  =  6000 
lb.  ft. ;  1  ft.  from  the  wall  it  is  400  x  14  =  5600  lb.  ft.,  and  so  on. 

22.  A  beam  projects  horizontally  15  ft.  from  a  vertical  wall. 
Construct  a  curve  to  show  the  relation  between  the  distance 
and  the  weight  if  the  bending  moment  at  the  wall  is  kept  at 
1200  lb.  ft. 


CHAPTER  V 

SPECIFIC  GRAVITY 

23.  Mass.  The  mass  of  a  body  is  the  quantity  of  matter 
(material)  contained  in  it.  The  English  unit  of  mass  is  a  cer- 
tain piece  of  platinum  kept  in  the  Exchequer  Office  in  London. 
This  lump  of  platinum  is  kept  as  a  standard  and  is  called  a 
pound.  The  metric  unit  of  mass  is  a  gram;  it  is  the  mass  of 
a  cubic  centimeter  of  distilled  water  at  4°  C.  (39.2°  F.). 

24.  Weight.  The  weight  of  a  body  is  the  force  with  which 
the  earth  attracts  it.  The  mass  of  a  pound  weight  would  not 
change  if  it  were  taken  to  different  places  on  the  surface  of  the 
earth,  but  its  weight  would  change.  A  piece  of  brass  which 
weighs  a  pound  in  Chicago  would  weigh  a  little  more  than  a 
pound  at  the  north  pole  and  a  little  less  than  a  pound  at  the 
equator.  Why  ?  The  masses  of  two  bodies  are  usually  com- 
pared by  comparing  their  weights. 

25.  Density.  The  density  of  a  body  is  the  quantity  of  mat- 
ter in  a  unit  volume.  Thus  with  the  foot  and  pound  as  units 
the  density  of  water  at  60°  F.  is  about  62.4,  since  1  cu.  ft.  of 
water  at  60°  F.  weighs  about  62.4  lb.  In  metric  units  the 
density  of  water  at  4°  C.  is  1,  since  1  ccm.  of  water  at  4°  C. 
weighs  1  g.  The  density  of  lead  in  English  units  is  707 ;  that 
is,  1  cu.  ft.  of  lead  weighs  707  lb.  In  metric  units  the  density 
of  lead  is  11.33,  since  1  ccm.  of  lead  weighs  11.33  g. 

26.  Specific  gravity.  The  specific  gravity  or  relative  density 
of  a  substance  is  the  ratio  of  the  weight  of  a  given  volume  of 
the  substance  to  the  weight  of  an  equal  volume  of  water  at 
4°  C.  (39.2°  F.).    Thus  if  a  cubic  inch  of  copper  weighs  .321  lb. 

42 


SPECIFIC  GRAVITY 


43 


and  a  cubic  inch  of  water  weighs  .0361  lb.,  the  specific  gravity 
of  this  piece  of  copper  is  .321  -r-  .0361  =  8.88.  If  we  are  told 
that  the  specific  gravity  of  silver  is  10.47,  it  means  that  a  cubic 
foot  of  silver  weighs  10.47  times  as  much  as  a  cubic  foot  of 
water. 

Approximate  Spkcific  Gravities 


Aluminum  .     . 

2.67 

Ice      .... 

.917 

Oak,  white 

.77 

Brass      .     .     . 

7.82 

Iron,  cast     .     . 

7.21 

Pine,  white    . 

.65 

Copper  .     .     . 

8,79 

Iron,  wrought  . 

7.78 

Pine,  yellow  . 

.66 

Cork       .     .     . 

.24 

Lead  .... 

11.3 

Silver    .     .     . 

10.47 

Glass,  white     . 

2.9 

Marble    .     .     . 

2.7 

Steel     .     .     . 

7.92 

Granite  .     .     . 

2.6 

Mercury,  at  60° 

13.6 

Tin  ...     . 

7.29 

Gold  .... 

19.26 

Nickel     .     .     . 

8.8 

Zinc      .     .     . 

7.19 

Exercise.  Find  the  specific  gravity  of  several  blocks  of  wood 
and  pieces  of  metal. 

Problem.  The  dimensions  of  a  block  of  cast  iron  are  3^  in. 
by  2f  in.  by  1  in.,  and  it  weighs  37.5  oz.  Find  its  specific  gravity. 

3^  X  2f  X  1  =  8.94  cu.  in. 
1  cu.  in.  of  water  =  .0361  lb. 
8.94  cu.  in.  of  water  =  .0361  x  16  x  8.94  oz. 
=  5.15  oz. 
AA'eight  of  block  of  metal 


Sp.  gr. 


Weight  of  equal  volume  of  water 
_37.5 
~5.15 
=  7.28. 


PROBLEMS 

1.  What  is  the  weight  of  1  cu.  in.  of  copper  ?                    ^  jpt 

Solution.   1  cu  in.  of  water  =  .0361  lb.  -0361 

Specific  gravity  of  copper  is  8.79 ;  that  is,  copper  is  8.79       264 

times  as  heavy  as  water.  52 

.-.  1  cu.  in.  of  copper  =  .0361  x  8.79  lb.  1 

=  .317  lb.  .317 


44  APPLIED  MATHEMATICS 

2.  What  is  the  weight  of  1  eu.  ft.  of  cast  iron  ?  ^^.4 

Solution.  1  cu.  ft.  of  water  =  62.4  lb.  '^•^^ 

Specific  gravity  of  cast  iron  is  7.21.  437 

.-.  1  cu.  ft.  of  cast  iron  =  62.4  x  7.21  lb.  12 

=  450  lb.  _J; 

450 

3.  Find  the  weight  of  1  cu.  in.  of  (a)  aluminum ;  (b)  cork ; 
(c)  lead;  (d)  gold;  (e)  silver;  (/)  zinc. 

4.  Find  the  weight  of  1  cu.  ft.  of  (a)  granite ;  (b)  ice ; 
(c)  marble ;  (d)  white  oak ;  (e)  yellow  pine. 

5.  What  is  the  weight  of  a  yellow-pine  beam  20  ft.  long, 
8  in.  wide,  and  10  in.  deep  ? 

6.  The  ice  box  in  a  refrigerator  is  24  in.  by  16  in.  by  10  in. 
How  many  pounds  of  ice  will  it  hold  ? 

7.  A  piece  of  copper  in  the  form  of  an  ordinary  brick  is 
8  in,  by  4  in.  by  2  in.  What  is  its  weight  ?  How  much  would 
a  gold  brick  of  the  same  size  weigh  ? 

8.  A  flask  contains  12  cu.  in.  of  mercury.  Find  the  weight 
of  the  mercury. 

9.  Find  the  weight  of  a  gallon  of  water. 

10.  What  is  the  weight  of  a  quart  of  milk  if  its  specific 
gravity  is  1.03  ? 

11.  How  many  cubic  inches  are  there  in  a  pound  of  water  ? 
Solution.  1  cu.  in.  =  .0361  lb. 

.-.  1  lb.  = cu.  in. 

■    .0361 

=  27.7  cu.  in. 

12.  An  iron  casting  weighs  50  lb.    Find  its  volume. 

Solution.    Let         x  =  number  of  cubic  inches,  in  the  casting. 
.0361 X  =  weight  of  x  cu.  in.  of  water. 
7.21  X  .0361  X  =  weight  of  x  cu.  in.  of  cast  iron. 
50 
7.21  X  .0361 
=  192  cu.  in. 


SPECIFIC  GRAVITY  45 

13.  What  is  the  volume  of  50  lb.  of  aluminum  ? 

14.  How  many  cubic  feet  are  there  in  50  Vo.  of  cork  ? 

15.  How  many  cubic  inches  are  there  in  a  flask  which  just 
holds  6  lb.  of  mercury  ? 

16.  A  cubic  foot  of  bronze  weighs  552  lb.  What  is  its  spe- 
cific gravity  ? 

17.  Find  the  specific  gravity  of  a  block  of  limestone  if  a 
cubic  foot  weighs  182  lb. 

18.  A  cubic  inch  of  platinum  weighs  .776  lb.  What  is  its 
specific  gravity  ? 

19.  A  cedar  block  is  5  in.  by  3  in.  by  2  in.  and  weighs  10.5  oz. 
Find  its  specific  gravity. 

20.  .0928  cu.  ft.  of  metal  weighs  112  lb.  Find  its  specific 
gravity. 

21.  Each  edge  of  a  cubical  block  of  metal  is  2  ft.  If  it  weighs 
4450  lb.,  what  is  its  specific  gravity  ? 

22.  A  metal  cylinder  is  15.3  in.  long  and  the  radius  of  a 
cross  section  is  3  in.  If  it  weighs  176.6  lb.,  what  is  its  specific 
gravity  ? 

23.  The  specific  gravity  of  petroleum  is  about  .8.  How  many 
gallons  of  petroleum  can  be  carried  in  a  tank  car  whose  capacity 
is  45,000  lb.  ? 

27.  Advantage  of  the  metric  system.  So  far  we  have  been 
using  the  English  system,  and  we  have  had  to  remember  that 
1  cu.  in.  of  water  weighs  .0361  lb.  But  in  the  metric  system 
the  weight  of  1  ccm.  of  water  is  taken  as  the  unit  of  weight  and 
is  called  a  gram.  Thus  8  ccm.  of  water  weighs  8  g.  If  a  cubic 
centimeter  of  lead  weighs  11.33  g.,  it  is  11.33  times  as  heavy 
as  water ;  hence  its  specific  gravity  is  11.33.  The  weight  in 
grams  of  a  cubic  centimeter  of  any  substance  is  its  specific  gravity. 

Exercise.   To  show  that  1  ccm.  of  water  weighs  1  g. 
Balance  a  glass  graduate  on  the  scales.    Pour  into  it  10,  20,  30  ccm. 
of  water,  and  it  will  be  found  that  the  weight  is  10,  20,  30  g. 


46  .  APPLIED  MATHEMATICS 

What  is  the  weight  of  80  ccm.  of  water  ?  A  dish  8  cm.  by 
5  cm.  by  2  cm.  is  full  of  water ;  how  many  grams  does  the 
water  weigh  ?  A  block  of  wood  is  12  cm.  by  10  cm.  by  5  cm.; 
what  is  the  weight  of  an  equal  volume  of  water  ?  A  brass 
cylinder  contains  125  ccm. ;  what  is  the  weight  of  an  equal  vol- 
ume of  water  ?  Hence  the  volume  of  a  body  in  cubic  centi- 
meters is  equal  to  the  weight  in  grams  of  an  equal  volume 
of  water. 

28.  First  method  of  finding  specific  gravity. 

1.  Weigh  the  solid  in  grams. 

2.  Pind  the  volume  of  the  solid  in  cubic  centimeters. 

1  ccm.  of  water  =  1  g. 
.'.  the  volume  in  cubic  centimeters  equals  the  weight  of  an 
equal  volume  of  water. 

Weight  in  grams 


3. 


Weight  of  an  equal  volume  of  water 
Weight  in  grams 


Volume  in  cubic  centimeters 


=  Sp.  gr. 


Exercise.  Find  the  specific  gravity  of  (a)  a  brass  cylinder; 
(h)  a  brass  prism ;  (c)  a  steel  ball ;  (d)  a  copper  wire ;  (e)  an 
iron  wire ;  (/)  a  pine  block ;  (g)  a  piece  of  oak.  Can  you 
expect  to  obtain  the  specific  gravities  given  in  the  table  ? 
Why  not? 

PROBLEMS 

1.  A  block  of  metal  13.8  cm.  by  14.2  cn\.  by  27.0  cm.  weighs 
60  kg.    Find  its  specific  gravity. 

2.  A  cylinder  is  84.3  mm.  long  and  the  radius  of  its  base  is 
15.4  mm.    If  it  weighs  157  g.,  what  is  its  specific  gravity  ? 

3.  A  metal  ball  of  radius  21.5  mm.  weighs  292.6  g.  Find  its 
"Specific  gravity. 

4.  The  altitude  of  a  cone  is  42.1  mm.  and  the  radius  of  the 
■)ase  is  14.6  mm.    Find  its  specific  gravity  if  it  weighs  22.3  g. 


SPECIFIC  GRAVITY 


47 


5.  How  many  times  heavier  is  (a)  gold  than  silver  ?  (b) 
gold  than  aluminum  ?  (c)  mercury  than  copper  ?  (rf)  steel 
than  aluminum  ?  (e)  platinum  than  gold  ?  (/)  cork  than  lead  ? 

6.  The  pine  pattern  from  which  an  iron  casting  is  made 
weighs  15  lb.  About  how  much  will  the  casting  weigh  ?  (The 
usual  foundry  practice  is  to  call  the  ratio  16  : 1.) 

29.  The  principle  of  Archimedes.  This  principle  furnishes  a 
convenient  method  of  finding  the  specific  gravity  of  substances. 

Exercise.  Weigh  a  brass  cylinder ;  weigh  it  when  suspended 
in  water  and  find  the  difference  of  the  weights.  Lower  the 
cylinder  into  an  overflow  can  filled  with 
water  and  catch  the  water  in  a  beaker 
as  it  flows  out.  Compare  the  weight 
of  this  water  with  the  difference  in  the 
weights.  Do  this  with  several  pieces 
of  metal.    What  seems  to  be  true  ? 

Imagine  a  steel  ball  submerged  in 
water  resting  on  a  shelf.  If  the  shelf 
were  taken  away,  the  ball  would  sink  to  the  bottom  of  the  tank. 
Now  suppose  the  surface  of  the  ball  contained  water  instead 
of  steel,  and  suppose  the  inclosed  water  weighed  5  oz.  If  the 
shelf  were  removed,  the  water  ball  would  be  held  in  its  posi- 
tion by  the  surrounding  water ;  that  is,  when  the  steel  ball  is 
suspended  in  water,  the  water  holds  up  5  oz.  of  the  total  weight 
of  the  ball. 

Principle  of  Akchimedes.  Any  body  when  suspended  in 
water  loses  in  weight  an  amount  equal  to  the  weight  of  its  own 
volume  of  water. 

30.  Second  method  of  finding  specific  gravity. 

1.  Weigh  a  piece  of  cast  iron,  156.3  g. 

2.  Weigh  it  when  suspended  in  water,    134.3  g. 

3.  156.3  -  134.3  =  22.0  g.  ^his  is  the  weight  of  an  equal 
volume  of  water. 


Fig.  19 


48  APPLIED  MATHEMATICS 

.    o,  156.3      _  .  - 

4.  Sp.gi-.  =  ^2:^  =  7.10. 

Let  W  =  the  weight  of  the  substance  in  air. 

to  =  the  weight  of  the  substance  suspended  in  water. 

W 

Then =  the  specific  gravity  of  the  substance. 

W  —  10 

Exercise.  Find  by  this  method  the  specific  gravity  of  (a)  brass ; 
(0)  copper ;  (c)  cast  iron ;  (d)  glass ;  (e)  lead ;  (/)  porcelain ; 
(g)  an  arc-light  carbon. 

PROBLEMS 

1.  How  much  will  a  brass  50-g.  weight  weigh  in  water  ? 

Solution.    Let  x  =  the  weight  in  water. 

-^^  =  7.82. 
50  -  X 

Solving,  X  =  43.6  g.    Check  by  experiment. 

2.  Compute  the  weight  in  water  of  (a)  100  g.  of  copper ; 

(b)  500  g.  of  zinc;   (c)   1kg.  of  silver;   (d)   200  g.   of  pine; 
(e)  100  g.  of  cork. 

3.  Find  the  weight  in  water  of  (a)  1  lb.  of  cast  iron  ;  (b)  1  lb. 
of  lead ;  (c)  5  lb.  of  aluminum ;  (d)  1  T.  of  granite ;  (e)  10  lb. 
of  cork. 

4.  If  a  boy  can  lift  150  lb.,  how  many  pounds  of  the  follow- 
ing substances  can  he  lift  under  water :  (a)  platinum  ?  (b)  lead  ? 

(c)  cast  iron  ?  (d)  aluminum  ?  (e)  granite  ? 

Solution,    (a)  The  problem  is  to  find  the  weight  in  air  of  a 

mass  of  platinum  which  weighs  150  lb.  in  water. 

Let  10  =  the  weight  in  air. 

w 
— -  =  22  (specific  gravity  of  platinum). 

Solving,  to  =  157  lb. 

5.  Construct  a  curve  to  show  the  weight  in  air  of  masses 
which  weigh  1  lb.  in  water,  th^  specific  gravity  varying  from 
1  to  20. 


SPECIFIC  GRAVITY  49 

6.  A  coppei'  cylinder  weighs  80  lb.  under  water.  How  much 
does  it  weigh  in  air  ? 

7.  A  cake  of  ice  just  floats  a  boy  weighing  96  lb.  How  many 
cubic  feet  are  there  in  it  ? 

Suggestion.  1  cu.  ft.  of  water  weighs  62.4  lb.  How  much  does 
1  cu.  ft.  of  ice  weigh?  How  many  pounds  will  1  cu.  ft.  of  ice  float? 
How  many  cubic  feet  of  ice  are  required  to  float  96  lb.  ? 

8.  A  pine  beam  1  ft.  square  is  floating  in  water.  If  its  spe- 
cific gravity  is  .55,  how  long  must  it  be  to  support  a  man 
weighing  180  lb.  ? 

9.  Construct  a  graph  to  show  the  weight  in  water  of  masses 
of  cast  iron  weighing  from  1  to  100  lb.  in  air,  given  that  the 
specific  gravity  of  cast  iron  is  7.2.  Why  should  the  graph  be 
a  straight  line  ? 

MISCELLANEOUS  PROBLEMS 

1.  Find  the  weight  of  50  ccm.  of  copper. 

Solution.  1  ccm.  of  water  =  1  g. 

Specific  gravity  of  copper  =  8.79. 
.-.  Weight  of  50  com.  of  copper  =  50  x  8.79  g. 

=  440  g. 

2.  Find  the  weight  of  (a)  100  ccm.  of  mercury  ;  (b)  150  ccm. 
of  zinc ;  (c)  300  ccm.  of  aluminum. 

3.  Find  the  volume  of  300  g.  of  zinc. 

Solution.        1  g.  of  water  has  a  volume  of  1  ccm. 

Specific  gravity  of  zinc  =  7.19. 

.-.  7.19  g.  of  zinc  has  a  volume  of  1  ccm. 

300       .,  ^ 

— —  =  41.7  ccm. 

7.19 

4.  Find  the  volume  of  (a)  50  g.  of  brass ;  (b)  100  g.  of  cork ; 
(c)  100  g.  of  gold ;  (d)  150  g.  of  marble ;  (e)  1  kg.  of  silver. 

5.  The  dimensions  of  a  rectangular  maple  block  are  8.1  cm., 
5.2  cm.,  and  3.5  cm.   If  it  weighs  100  g.,  find  its  specific  gravity. 


50  APPLIED  MATHEMATICS 

6.  109  ccm.  of  copper  and  34  ccm,  of  zinc  are  melted  to- 
gether to  form  brass.    Find  its  specific  gravity. 

Solution.   Let  s  =  the  specific  gravity  of  the  brass. 

109  +  34  =  143  ccm.,  volume  of  the  brass. 
143  s  =  weight  of  the  brass. 
109  X  8.79  =  weight  of  the  copper. 
34  X  7.19  =  weight  of  the  zinc. 

143  s  =  109  X  8.79  +  34  x  7.19. 
Solve  for  s  and  check. 

7.  58.8  g.  of  copper  and  25.2  g.  of  zinc  are  combined  to  form 
brass.    What  is  its  specific  gravity  ? 

Solution.    Let  s  =  specific  gravity  of  the  brass. 

58.8  +  2.5.2  =  84  g.,  weight  of  the  brass. 

—  =  volume  of  the  brass. 

— '—  =  6.69  =  volume  of  the  copper. 
8.79  ^^ 

25.2 

— '—  =  3.50  =  volume  of  the  zinc. 

7.19 

84 

—  =  6.69  +  3.50. 
s 

Solve  for  s  and  check. 

8.  The  specific  gravity  of  a  piece  of  brass  weighing  123  g.  is 
8.22.    How  many  grams  of  copper  and  of  zinc  are  there  in  it  ? 

Solution.   Let  c  =  number  of  grams  of  copper. 

z  =  number  of  grams  of  zinc. 

c 

=  volume  of  the  copper. 

8.79  ^^ 

z 
——-  =  volume  of  the  zinc. 
7.19 

123 

=  volume  of  the  brass. 

8.22 

c  +  z  =  123. 

c  z     ^  123 

8.79      7.19  ~  8.22  ' 

Solve  and  check. 


SPECIFIC  GRAVITY  51 

9.  An  alloy  was  formed  of  79.7  ccm.  of  copper  and  51.4  ccm. 
of  tin.    Find  its  specific  gravity. 

10.  475.2  kg.  of  hard  gun  metal  was  made  by  combining 
411  kg.  of  copper  and  64.2  kg.  of  tin.  What  was  its  specific 
gravity  ? 

11.  336  lb.  of  copper  and  63  lb.  of  zinc  were  combined  to 
make  brazing  metal.    Find  its  specific  gravity. 

Suggestion.  To  reduce  pounds  to  grams  multiply  by  4.53.6.  Since 
this  factor  occurs  in  each  term  of  the  equation,  it  may  be  divided  out. 

12.  Nickel-aluminum  consists  of  20  parts  of  nickel  and  80 
parts  of  aluminum.    Find  its  specific  gravity. 

13.  What  is  the  specific  gravity  of  bell  metal  consisting  of 
SO  per  cent  copper  and  20  per  cent  tin  ? 

14.  Find  the  specific  gravity  of  Tobin  bronze,  which  consists 
of  58.22  per  cent  copper,  2.30  per  cent  tin,  and  39.48  per  cent 
zinc. 

15.  516  g.  of  copper,  258  g.  of  nickel,  and  226  g.  of  tin  are 
combined  to  form  German  silver.    Find  its  specific  gravity. 

16.  How  much  copper  and  how  much  aluminum  must  be 
taken  to  make  200  kg.  of  aluminum  bronze  having  a  specific 
gravity  of  7.69  ? 

17.  A  mass  of  gold  and  quartz  weighs  500  g.  The  specific 
gravity  of  the.  mass  is  6.51  and  of  the  quartz  is  2.15.  How 
many  grams  of  gold  are  there  in  the  mass  ? 


CHAPTER  VI 

GEOMETRICAL  CONSTRUCTIONS  WITH  ALGEBRAIC 
APPLICATIONS 

Note.  Make  all  drawings  and  constructions  in  a  notebook. 
Record  all  the  work  in  full,  having  it  arranged  neatly  on  the  page. 
Make  the  constructions  as  accurately  as  possible. 

31.  Drawing  straight  lines.  Keep  the  pencil  sharp,  and 
make  the  lines  heavy  enough  to  be  clearly  seen. 

Exercise  1.    Draw  a  line  2  in.  long. 

d ^ P 

Fig.  20 

To  do  so  most  accurately,  draw  an  indefinite  line  AB.  Then  put 
your  compasses  on  the  scale  of  the  ruler  so  that  the  points  are  2  in. 
apart.   With  yl  as  a  center  strike  an  arc  at  C  AC  is  the  required  line. 

Exercise  2.  Using  this  method,  draw  lines  as  follows : 
(a)  If  in. ;  (h)  1  dm. ;  (c)  1  cm. ;  (d)  83  mm. ;  (e)  3.5  cm. ; 
(/)  136  mm. 

32.  Drawing  to  scale.  Choose  a  scale  that  will  give  a  good- 
sized  figure,  and  below  every  figure  record  the  scale  used. 

Exercise  3.  The  distance  between  two  towns  A  and  5  is  30 
mi.  How  could  a  line  6  cm.  long  represent  that  distance  ? 
Draw  such  a  line  and  explain  the  relation  that  exists  between 
the  distance  and  the  line. 

Exercise  4.  Draw  a  line  3  in.  long  and  let  it  represent  a  dis- 
tance of  36  mi.  What  distance  is  represented  by  1  in.  ?  by 
2  in.  ?  by  1\  in.  ?  by  2%  in.  ?  In  this  exercise  the  distance  is 
said  to  be  represented  on  a  scale  of  1  in.  to  12  mi. 

52 


GEOMETRICAL  CONSTRUCTIONS  53 

Exercise  S.  With  a  scale  of  1  in.  to  16  ft.  (1  in.  =  16  ft.) 
draw  lines  to  represent  the  distances  (a)  8  ft. ;  (b)  12  ft. ; 
(c)  24  ft. ;  ((/)  36  ft. ;  (e)  18  ft. 

33.  Measuring  straight  lines.  With  an  unmarked  ruler  or 
with  the  edge  of  your  book  draw  a  line  AJ3.  To  locate  the  ends 
of  the  line  as  accurately  as  possible,  make  small  marks  in  the 
paper  at  A  and  B  with  the  point  of  the  compasses.  Care  should 
be  taken  that  the  marks  do  not  penetrate  to  the  surface  below. 
Place  one  point  of  the  compasses  at  A  and  let  the  other  fall  at 
B.  With  this  opening  of  the  compasses  place  the  points  against 
the  scale  of  a  ruler,  one  point  on  the  division  marked  1  cm., 
and  count  the  number  of  centimeters  and  tenths  of  a  centi- 
meter between  the  points  of  the  compasses.  On  the  line  AB 
write  its  length  as  you  have  found  it.  (The  end  divisions 
of  a  ruler  are  not  usually  so  accurate  as  the  middle  divisions ; 
hence  in  making  a  measurement  it  is  best  not  to  start  at  the 
zero  of  the  scale.) 

Exercise  6.  Make  two  crosses  in  your  notebook  and  call  the 
points  of  intersection  M  and  N.  Using  the  compasses,  measure 
MN  in  inches  and  centimeters  and  record  the  result. 

Exercise  7.  Draw  an  indefinite  lipe  AX  and  mark  off  on  it 
^fi  =  2.8  cm.,  BC  =  1.7  cm.,  and  CD  =  3.4  cm.  Then  with 
your  compasses  measure  AD.  Record  the  length  and  compare 
it  with  the  sum  of  the  numbers. 

Exercise  8.  (a)  Measure  the  lines  AB,  CD,  and  EF,  Eecord 
the  measurements  and  add  them. 

^ PQ. p  ^ r 


Fig.  21 


(h)  Draw  an  indefinite  line  AX  and  mark  off  on  it  AB,  CD, 
and  EF,  the  point  C  falling  on  B  and  the  point  E  on  D. 
Measure  AF  and  record  the  result.  Compare  with  that  ob- 
tained in  (a). 


64  APPLIED  MATHEMATICS 

34.  Angles.  An  angle  is  formed  by  two  lines  that  meet. 
Thus  the  lines  BC  and  BA  meet  at  the  vertex  B,  forming  the 
angle  ABC,  B,  or  m.  When  three  letters 
are  used  to  denote  an  angle  the  letter 
at  the  vertex  is  read  between  the  other 
two.    The  single  small  letter  should  be  Fig.  22 

used  to  denote  an  angle  when  convenient 

The  size  of  an  angle  depends  on  the  amount  of  opening  be- 
tween the  lines. 

A  right  angle  is  an  angle 
of  90°. 

An  acute  angle  is  less  than 
90°.  ^"'-  ^ 

An  obtuse  angle  is  greater  than  90°  and  less  than  180°. 

Thus  a  is  an  acute  angle  and  b  is  an  obtuse  angle. 

35.  The  protractor.  To  measure  an  angle  place  the  pro- 
tractor so  that  the  center  of  the  graduated  circle  is  at  the  ver- 
tex of  the  angle  and  its  straight  side  lies  along  one  arm  of 
the  angle.  Note  the  graduation  under  which  the  other  arm  of 
the  angle  passes. 

Exercise  9.  Take  a  piece  of  paper  and  fold  it  twice  so  that  the 
creases  will  form  four  right  angles  at  a  point.  Test  one  of  the 
angles  with  the  protractor. 

Exercise  10.  About  a  point  construct  angles  of  42°,  85°,  and 
53°.    What  is  the  test  of  accuracy  of  construction  ? 

Exercise  11.  At  each  end  of  a  line  AB,1  cm.  long,  construct 
an  angle  of  60°  so  that  AB  \h  one  arm  of  each  angle  and  the 
other  arms  intersect  at  C.  Measure  angle  A  CB,  and  write  the 
number  of  degrees  in  each  angle.  Measure  ^C  and  BC.  What 
is  the  test  of  accuracy  of  construction  ?  Bisect  angle  A  CB  by 
the  line  CD,  D  being  on  A  B.   How  much  longer  is  ^  C  than  AD? 

Exercise  12.  Draw  a  large  triangle.  Measure  each  angle  and 
write  the  results  in  the  angles.    What  ought  to  be  the  sum  ? 


GEOMETRICAL  CONSTRUCTIONS 


55 


Exercise  13.  Make  an  angle  A  =  37°.  On  the  horizontal  arm 
take  AC  =  &  cm.  and  on  the  other  arm  take  AB  =1.5  cm. 
Draw  BC.  Guess  the  number  of  degrees  in  angle  ACE.  Meas- 
ure it. 

Exercise  14.  To  find  the  distance  across  a  lake  from  A  to  B, 
a  surveyor  selected  a  point  C  from  which  he  could  see  both  A 
and  B.  He  measured  the  angle  A  CB,  72°,  with  a  transit  and' 
found  the  distances  CA  and  CB  to  be  450  ft.  and  400  ft.  re- 
spectively. From  these  measurements  draw  the  figure  to  scale ; 
measure  AB  and  determine  what  distance  it  represents. 

Exercise  15.  To  find  the  height  of  a  building  AB  across  a 
river  DB  measurements  were  made  as  follows :  angle  A  CB  = 
16°,  angle  A  DB  =  37°, 
and  CD  =  100  ft.  Draw 
to  scale,  and  find  the 
height  of  the  building 
and  the  width  of  the 
river. 


Fig.  24 


Exercise  16.  A  man  wishing  to  find  the  distance  between  two 
buoys,  A  and  B,  measured  a  base  line  CD  1500  ft.  in  length 
along  the  shore.  At  its  extremities,  C  and  D,  he  measured  the 
following  angles  :  angle  DCB  =  36°  15',  angle  BCA  =  50°  45', 
angle  CDA  =  43°  30',  and  angle  ADB  =  72°.  Draw  to  scale, 
and  find  the  distance  between  the  buoys. 

36.  From  a  point  in  a  line  to  draw  a  line  at  right  angles 
(perpendicular)  to  it. 

Construction.  Let  C  be  the  point  in  AB  from  which  the  line  is 
to  be  drawn.  Place  one  point  of  the  com-  F 

passes  at  C  and  mark  o&^onAB  the  equal  y  ^ 

distances  CD  and  CE.  With  D  and  E 
as  centers  and  a  convenient  radius  de- 
scribe arcs  intersecting  at  F.  Draw  CF. 
FCB  is  a  right  angle,  and  CF  is  said  to 
be  perpendicular  to  AB. 


A     D 


C 
Fig.  25 


3 


56 


APPLIED  MATHEMATICS 


Example.    To  construct  a  right  triangle  whose  legs  are  6  cm. 
and  8  cm.  respectively. 

Construction.  Draw  an  indefinite 
line  AX  and  mark  off  .1 C  =  8  cm.  At 
the  point  C  construct  the  perpendicular 
CY  and  take  CB  =  Q  cm.  Draw  AB, 
and  ABC  is  the  required  triangle. 

Measure  c  =  9.95  cm. 

Check  your  construction  by  the 
formula 

fj2  +  ifl  =  ^2 


Fig.  26 


where  a  and  h  are  the  legs  of  a  right  triangle  and  c  is  the 
hypotenuse. 

a2  +  ^2  ^  62  +  82 
=  36  +  64. 
c2  =  100. 
c2  =  9.952  ^  99,0.      - 


9.95 

896 

89 

5 

99.0 


Exercise  17.  Construct  to  scale  if  necessary  and  check  as  in 
the  preceding  exercise,  given  a  and  h.  (a)  3.5  cm.  and  6.8  cm. ; 
(h)  4.3  cm.  and  9.6  cm. ;  (c)  84  mm.  and  64  mm. ;  (d)  42  in.  and 
18  in. ;  (e)  28  ft.  and  16  ft. ;  (/)  120  mi.  and  200  mi. 

Exercise  18.  Construct  a  square 
whose  side  is  4  cm.  D  c 

Construction.  Make  ^B  =  4  cm. 
At  B  draw  BX  perpendicular  to  A  B. 
Cut  off  £C  =  4  cm.  With  A  and  C 
as  centers  and  a  radius  of  4  cm.  draw 
arcs  intersecting  at  D.  Draw  A  D  and 
CD.  A  BCD  is  the  required  square. 
Measure  the  diagonal  and  record  the 
result  on  the  figure.  Check  by  apply- 
ing the  formula  of  the  right  triangle. 


Fig.  27 


Exercise  19.    Construct   to  scale  squares  whose    sides    are 
(a)  12  in. ;  (b)  1.8  m. ;  (c)  540  mm.    Check  by  formula. 


i 


2)     S 


GEOMETRICAL  CONSTRUCTIONS  57 

Exercise  20.  Construct  to  scale  and  check,  rectangles  whose 
sides  are  (a)  78  and  48  cm. ;  (h)  32  and  54  in. ;  (c)  482  and 
615  ft. 

37.  To  construct  a  perpendicular  to  a  line  from  a  point 
outside  the  line. 

Let  AB  be  the  line  and  C  the  point. 
With  C  as  a  center  describe  an  arc 

cutting  AB  dX  D  and  E.    With  D  and       y  I ^ 

E  as  centers  and  a  convenient  radius 
describe  arcs  intersecting  at  F.  Draw 
CF,  the  required  perpendicular.  Fig.  28 

Exercise  21.  Construct  right  triangles  whose  legs  are  {a)  6  and 
12  cm. ;  (&)  5  and  9  cm.  Draw  perpendiculars  from  the  vertex 
of  the  right  angle  to  the  hypotenuse.    Measure  and  check. 

Exercise  22.  Draw  a  large  triangle  and  construct  a  perpen- 
dicular from  the  vertex  to  the  base.  Measure  the  sides  of  the 
two  right  triangles  formed  and  check  by  the  formula. 

38.  To  construct  a  triangle  whose  sides  are  given. 

Exercise  23.  Construct  a  triangle  whose  sides  are  7,  8,  and 

10  cm.  respectively. 

Construction.  Draw  a  line  AB  10  cm.  long.  With  yl  as  a  center 
and  a  radius  of  7  cm.  describe  an  arc.  With  £  as  a  center  and  a 
radius  of  8  cm.  describe  an  arc  cutting  the  first  arc  at  C  Draw  A  C 
and  BC,  and  ABC  is  the  required  triangle. 

Exercise  24.  Erom  C  in  the  figure  of  Exercise  23  draw  a  per- 
pendicular to  AB.  Measure  the  sides  of  the  right  triangles  and 
check  by  the  formula. 

Exercise  25.  Construct  a  triangle  whose  sides  are  7.5,  8.5,  and 

11  cm.  respectively.  Draw  a  perpendicular  from  the  vertex  to 
the  base  and  find  the  area  of  the  triangle.  Check  by  drawing  a 
perpendicular  to  another  side  and  use  its  length  to  find  the  area. 
The  perpendicular  from  the  vertex  to  the  base  is  called  the 
altitude  of  the  triangle. 


58 


APPLIED  MATHEMATICS 


t 


i 


D 

Fig.  29 


39.  To  bisect  a  given  line. 

Exercise  26.  Bisect  a  given  line  AB. 

Construction.  With  A  and  B  as  cen- 
ters and  a  convenient  radius  describe  arcs 
intersecting  at  C  and  D.  Draw  CZ),  inter- 
secting AB2XE.  Then  AE  =  EB.  Check 
by  measuring. 

Exercise  27.  Draw  an  indefinite  line 
AB  and  divide  it  into  four  equal  parts,  using  the  method  of 
arcs.    Check. 

Exercise  28.  Construct  an  equilat- 
eral triangle  ABC  whose  sides  are 
each  9  cm.  Divide  the  base  into  four 
equal  parts.  Draw  CD  and  CF  and 
measure  their  lengths.  Measure  the 
angle  ADC.  Applying  the  formula 
of  the  right  triangle,  compute  CD 
and  CF. 


Fig.  30 


40.  To  bisect  an  angle. 


Exercise  29.  Make  an  angle  BA  C  and  bisect  it. 

Construction.  With  A  as  a  center  and  with  a  rather  large  radius 
mark  two  points  D  and  E  on  AC  and 
AB  respectively.  With  D  and  E  as  cen- 
ters and  the  same  radius  describe  arcs 
intersecting  at  F.  Draw  AF,  and  angle 
£.4F=  angle  FAC.  Check  with  tiie 
protractor. 

Exercise  30.  Draw  an  obtuse  angle 
and  bisect  it.    Check. 

Exercise  31.  Construct  a  triangle  ABC  with  AB  =  7.6  cm., 
AC  =  6.5  cm.,  and  angle  A  =  45°.  Construct  the  altitude  CD 
and  measure  its  length.  Check  by  computing  the  length  of  CD, 
using  the  formula  of  the  right  triangle. 


Fig.  31 


GEOMETRICAL  CONSTRUCTIONS  59 

41.  Parallel  lines.  Lines  that  lie  in  the  same  plane  and  do 
not  meet  however  far  produced  are  called  parallel  lines. 

Exercise  32.  Construct  a  rectangle  whose  dimensions  are  4.35 
and  7.85  cm.  respectively.  Find  the  area  to  three  significant 
figures.  The  opposite  sides  of  a  rectangle  are  parallel.  Write 
in  your  notebook  the  sides  that  are  parallel. 

42.  Parallelograms.  If  the  opposite  sides  of  a  four-sided 
figure  are  parallel,  the  figure  is  called 

a  parallelogram.   ABCD  is  a  paral- 
lelogram. 

Exercise  33.    Construct   a   paral- 
lelogram, with   AB  =  ^  cm.,  AD  =  Fig.  32 
5  cm.,  and  angle  A  —  65°.  The  point 

C  can  be  obtained  with  arcs,  as  in  Exercise  18.   Name  the  par- 
allel sides.    Measure  all  the  angles. 

Exercise  34.  Construct  a  parallelogram  with  ^ii  =  9.45  cm., 
BC  =  4.15  cm.,  and  angle  B  =  115°.  From  D  construct  DE 
perpendicular  to  AB,  E  being  on  AB.  The  line  DE  is  the  alti- 
tude of  the  parallelogram.  Measure  DE  and  find  the  area  of 
the  parallelogram. 

43.  To  draw  a  line  parallel  to  a  given  line. 

Exercise  35.  Construct  a  triangle  with  A B  —  8  cm.,  BC  =  ^  cm., 

and  /I  C  =  6  cm.    Take  CD  =  4  cm. 

Through  D  draw  DE    parallel   to 

AB.    (Construct  the  parallelogram 

ADFG.)    Measure   CE,   or  y,   and 

record    its    length.     The    equation 

4  y 

—  =  -^ —  will  give  the  length  of 

■"     "     y 

CE.    Solve  the  equation  and  com- 
pare with  the  measured  length. 

Exercise  36.  Construct  a  triangle  ABC  whose  sides  are: 
4-5  =  7  cm.,  BC  =  9  cm.,   and    CA  =  11  cm.     On   BC   take 


60  APPLIED  MATHEMATICS 

BD  =  3  cm.,  and  through  D  draw  a  parallel  to  AB.  Measure 
the  lengths  of  the  two  parts  oi  AC  and  check  by  an  equation 
like  that  in  Exercise  35. 

44.  To  construct  an  angle  equal  to  a  given  angle. 

Exercise  37.  At  the  point  D  on  DE  to  construct  an  angle 
equal  to  angle  A. 

Construction.  With  yl  as  a  center  and  a  rather  large  radius  de- 
scribe the  arc  BC  cutting  AX  at  B  and  A  Y  at  C.  AVith  Z)  as  a  cen- 
ter and  the  same  radius  describe  an  arc  FG  cutting  DE  at  F.   Take 


off  with  the  compasses  the  distance  BC;  then  with  F  as  a  center 
and  BC  as  a  radius  describe  an  arc  cutting  FG  at  H.  Draw  DH. 
Angle  D  is  the  required  angle  equal  to  A.   Check  with  the  protractor. 

Exercise  38.  Make  angles  of  {a)  40°,  {b)  58°,  (c)  140°,  and 
construct  angles  equal  to  them. 

Exercise  39.  Construct  a  triangle  ABC,  making  AB  =.  8.4  cm., 
BC  =  6.8  cm.,  and  AC  =  1.2  cm.  Draw  a  line  DE  =  4.2  cm. 
At  D  make  an  angle  EDF  equal  to  angle  BA  C,  and  at  E  make 
an  angle  DEF  equal  to  angle  ABC.  Produce  the  two  lines  till 
they  meet  at  F.  Measure  the  sides  and  angles  of  the  triangle 
DEF  and  compare  them  with  the  corresponding  parts  of  the 
triangle  ABC. 

Triangles  which  have  their  corresponding  angles  equal  and 
their  corresponding  sides  proportional  are  called  similar  tri- 
angles. 

Exercise  40.  The  angle  of  elevation  of  a  chiirch  steeple  at  a 
point  300  ft.  from  its  base  was  found  to  be  16°.   Construct  a 


GEOMETRICAL  CONSTRUCTIONS  61 

similar  triangle,  that  is,  draw  to  scale  and  find  the  height  of 
the  steeple. 

Exercise  41.  At  a  distance  of  500  ft.  the  angle  of  elevation  of 
the  top  of  one  of  the  "  big  trees  "  of  California  is  31°.  How  tall 
is  the  tree  ? 

Exercise  42.  Make  some  practical  problems  and  solve  them. 

PROBLEMS 

Record  all  measurements  and  give  the  work  in  full  in  your 
notebooks. 

1.  The  two  legs  of  a  right  triangle  are  15  and  36  ft.  respec- 
tively. Construct  the  triangle  to  scale,  stating  scale  used. 
Measure  the  hypotenuse.  Check  by  applying  the  formula  of 
the  right  triangle. 

2.  Construct  a  rectangle  4  cm.  by  7  cm.  Measure  the  diag- 
onal;   Check. 

3.  A  right  angle  may  be  constructed 
as  shown  in  Fig.  35.  ABC  is  an  equi- 
lateral triangle,  CD  =  BC.  AD  is  drawn 
and  BAD  is  a  right  angle.  Construct  a 
right  angle  DAB.  On  AB  take  AE  =  S.4: 
cm,,  and  on  AD  take  AF=:  3,5  cm.  Meas- 
ure EF.    Check. 

4.  The  hypotenuse  of  a  right  triangle 
is  19,4  ft,  and  one  leg  is  14.2  ft.    Com- 
pute the  length  of  the  other  leg.    Check  by  constructing  the 
triangle  to  scale  and  measuring  the  required  leg. 

5.  The  base  of  a  right  triangle  is  x,  the  altitude  is  x  ■}- 1, 
and  the  hypotenuse  is  a;  -f-  2,  Find  x  by  applying  the  formula 
of  the  right  triangle.  Check  by  constructing  a  right  triangle 
with  the  legs  x  and  x  -\- 1.  Measure  the  hypotenuse  and  com- 
pare with  the  value  oi  x  +  2. 

6.  The  following  sets  of  expressions  represent  the  sides  of 
a  right  triangle.    Solve  and  check  as  in  Problem  5. 


62  APPLIED  MATHEMATICS 

Legs  Hypotenuse 

(a)  X  and  x  +  3  x  +  6 

(ft)  a:  and  a;  +  7  x  +  8 

(g)  a;  and  a;  —  2  a:  +  2 

(J)  X  and  X  +  4  x  +  8 

(  e  )  X  and  x  —  7  x  +  1 

(/)  X  and  2  X  -  4  2  x  -  2 

(^)  X  and  X  +  1  2  x  —  11 

(A)  X  and  x  +  5  2  x  —  5 

7.  The  altitude  of  a  rectangle  is  1  ft.  less  than  the  base,  and 
the  area  is  20  sq.  ft.  Find  the  dimensions.  Check  by  drawing 
on  squared  paper  and  counting  the  squares. 

8.  The  following  sets  of  expressions  represent  the  sides 
and  the  area  of  a  rectangle.  Find  the  dimensions  and  check 
as  in  Problem  7. 

Sides  Area 

(a)  X  and  x  —  10  24 

(6)  X  and  X  —  7  30 

(c)  X  and  x  +  12  85    • 

\d)  X  and  x  +  9  ,90 

(e)  X  and  2x  +  5  18 

(/)  X  and  2  X  +  1  36 

\(l)  X  and  3  X  -  7  40 

(A)  X  and  4x-  10  24 

9.  Construct  a  right  triangle  ADC,  denoting  the  base  by  x 
and  the  altitude  by  y.  Complete  the  rectangle  xy.  How  is  the 
area  of  the  rectangle  found  ?  What  algebraic  expression  repre- 
sents it  in  this  case  ?  What  part  of  the  rectangle  is  the  triangle 
ABC  ?  What  algebraic  expression  represents  the  area  of  the 
triangle  ?  What  reason  can  you  give  for  the  correctness  of  the 
expression  for  the  area  of  the  triangle  ? 

10.  The  legs  of  a  right  triangle  are  x  and  x  +  6.  Its  area 
is  20.  Find  the  sides  of  the  triangle.  To  check,  draw  on 
squared  paper  a  right  triangle  whose  legs  are  x  and  x  -\-  6. 
Find  the  area  by  counting  the  large  squares  inside  the  triangle. 


GEOMETRICAL  CONSTRUCTIONS  63 

When  a  part  of  a  square  looks  less  than  a  half,  it  is  not 
counted;  but  if  it  looks  greater  than  a  half,  it  is  counted  as  a 
whole  square. 

11.  The  following  sets  of  expressions  represent  the  legs  and 
area  of  a  right  triangle.  Find  the  length  of  the  legs'  in  each 
case,  and  check  on  squared  paper  as  in  Problem  10. 

Legs  Area 

(a)  X  and  x  —  11  30 

(6)  a;  and  a;  -  12  14  ■ 

(c)  X  and  a;  +  10  28 

(d)  X  and  a;  -  15  27 

(e)  X  and  2x  —  7  15 
(/)  a:  and  5  a;  -  9  40 
(</)  X  and  3  a;  —  1  35 
(A)  a;  and  4  a;  —  9  45 

12.  Construct  a  parallelogram  ABCD.  Bisect  the  angles  A 
and  B,  and  let  the  bisectors  meet  at  F.  Measure  the  angle 
AFB.  Measure  AB,  BF,  and  FA.  Apply  the  formula  of  the 
right  triangle.  Make  the  test  in  several  parallelograms  and 
state  what  seems  to  be  true  of  the  bisectors  of  two  consecutive 
angles  of  a  parallelogram. 

13.  Construct  a  triangle  ABC  with  CB  =  8  cm.,  AB  =  10.5  cm., 

and  AC  =  5.5  cm.    On  CA  take  CD  =  2  cm.,  and  from  D  draw 

a  line  parallel  to  CB  intersecting  AB  at  E.    From  the  formula 

AD      AE 

——  =  ——  find  AE.    Check  by  measuring  AE. 

14.  In  the  figure  of  Problem  13  let  ^i)  =  ar,Z)C=  3,  ^S  =  a;+1, 
and  EB  =  5.  Use  the  formula  to  find  x.  Find  the  sides  A  C 
and  AB.  Check  by  construction,  taking  the  base  any  conven- 
ient length. 

15.  The  following  sets  of  values  are  the  segments  of  the 
sides  of  a  triangle  formed  by  a  line  parallel  to  the  base.  Find 
the  length  of  each  segment  and  check  by  constructing  the  tri- 
angle and  the  parallel  as  in  Problem  14. 


64  APPLIED  MATHEMATICS 


AD 

DC 

AE 

EB 

(a)  X 

3 

2-x 

2  +  x 

(b)    X 

2 

a;  +  5 

x-1 

(C)    X 

4 

x^-1 

x  +  7 

(d)    X 

5 

4-x 

3  +  x 

(e)  X 

3 

a;  +  2 

x  +  5 

(/)  ^ 

x  +  2 

X  +   4: 

2x-h 

(g)  X  x  +  3  2x—\  x 

(h)  X  x  +  4  Sx  —  2  x  +  5 

16.  The  legs  of  a  right  triangle  are  x  and  ?/,  and  their  sum 
is  15.  If  the  area  of  the  triangle  is  27,  find  x  and  ?/.  To  check 
the  result,  construct  on  squared  paper  a  right  triangle  whose 
legs  are  x  and  y.  Count  the  large  squares  and  compare  with 
the  given  area. 

17.  The  sum  of  the  legs  and  the  area  of  a  right  triangle  are 

given  by  the  following  sets  of  numbers.    Find  x  and  y,  and 

check. 

Legs  Sum  of  the  Legs  Area 

(a)  X  and  y  16  14 

(b)  X  and  y  25  42 

(c)  X  and  y  15  28 

(d)  X  and  y  19  36 

18.  The  difference  of  the  legs  of  a  right  triangle  and  the 
area  are  given  by  the  following  sets  of  nmnbers.  Find  x  and  y, 
and  check. 


Legs 

DiF 

ference  of 

THE  Legs 

Area 

(a)  X  and  y 

12 

32 

(b)  X  and  y 

10 

48 

(c)  X  and  y 

8 

64 

(d)  X  and  y 

9 

35 

(c)  X  and  y 

5 

102 

CHAPTER  VII 

THE  USE  OF  SQUARED  PAPER 

I.  Graphical  Representation  of  Tables  of  Values 

45.  The  results  of  experiments  and  observations,  statistical 
tables,  and  tabulated  numerical  data  of  all  kinds  can  be  repre- 
sented by  lines  and  curves.  The  graph  shows  at  a  glance  rela- 
tions which  are  not  so  evident  in  a  table  of  values ;  and  it  also 
enables  one  to  find  readily  values  which  lie  between  those 
given  in  the  table. 

Exercise.  Construct  a  graph  to  represent  this  record  of  tem- 
perature, given  in  The  Chicago  Daily  News. 


3  P.M 

4  P.M 

5  P.M 

6  P.M 

...  78 
...  77 
...  76 
...  75 

3  A.M 

4  A.M 

5  A.M 

6  A.M 

7  A.M 

....  73 
....  73 
....  72 
.  .  .    72 

7  P.M 

...  76 
...  75 
...  73 
...  74 
...  75 
...  73 
...  73 
...  73 

.  .  .  .  72 

8  P.M 

8  A.M 

.  .  .  .  71 

9  P.M 

9  A.M 

....  71 

10  P.M 

11  P.M 

10  A.M 

11  A.M 

....  75 
....  74 

12  midnight   .  .  . 

1  A.M 

2a.m 

12  noon  .... 

1  P.M 

....  75 
....  76 

We  have  here  two  quantities,  hours  and  degrees,  so  related 
that  to  a  change  in  one  there  is  a  corresponding  change  in  the 
other. 

66 


66 


APPLIED  MATHEMATICS 


The  sheets  of  squared  paper  we  use  have  seventeen  large 
squares  each  way ;  the  side  of  a  large  square  is  a  centimeter, 
and  of  a  small  square  a  millimeter. 

The  units  for  representing  an  hour  and  a  degree  should  be 
chosen  so  that  the  picture  may  be  of  good  size  and  still  allow 
the  whole  table  to  be  represented.  Let  the  horizontal  lines 
represent  time  and  the  vertical  lines  represent  temperature. 

lot- 


O        0        fr 


3 


r 


sen 


II  lAM     3 

Hme 

Fig.  36 


IRH 


The  horizontal  and  vertical  lines  from  which  we  count  degrees 
and  hours  are  called  axes.  We  will  always  mark  them  OX 
and  OY  respectively,  and  call  them  the  ic-axis  and  the  y-axis. 
The  point  O  is  called  the  origin. 

Since  the  number  of  degrees  is  always  greater  than  70,  we 
may  call  the  cc-axis  70°  to  save  space.  At  3  p.m.  the  tempera^ 
ture  is  78° ;  hence  on  the  3  o'clock  line  we  put  a  point  a»t  78°,  and 
so  on  for  the  other  hours,  as  shown  in  Fig.  36.  A  smooth  curve 
is  then  drawn  through  the  points,  and  we  have  a  curve  which 
shows  at  a  glance  the  change  in  temperature  during  the  day. 


THE  USE  OF  SQUARED  PAPER  67 

The  curve  does  not,  of  course,  show  the  exact  reading  of  the 
thermometer  between  the  hours.  However,  it  shows  when  the 
temperature  was  falling  and  when  rising,  whether  the  change 
was  rapid  or  gradual,  and  in  general  gives  a  fairly  correct 
representation  of  the  temperature  for  the  day. 

46.  Hints  on  the  use  of  squared  paper.  All  graphical  work 
should  be  done  in  a  book  of  squared  paper  where  it  can  be  re- 
ferred to  from  time  to  time.  Much  can  be  learned  by  looking 
back  over  the  curves  and  noting  the  relations  between  the 
various  problems  and  curves.  Frequently  it  will  be  found  that 
a  curve  of  the  same  shape  is  constructed  in  solving  several 
different  problems. 

Each  graphical  solution  ought  to  be  complete  in  itself.  The 
table  of  values  or  other  data  should  be  written  on  the  sheet 
with  the  curve,  or  on  the  blank  page  at  the  left  of  the  graph  in 
the  notebook.  The  axes  should  be  lettered  OX  and  OY,  and 
the  units  written  on  them.  It  is  not  necessary  that  the  units 
should  be  the  same  for  both  axes,  but  they  should  be  chosen 
so  that  the  whole  range  of  values  may  be  plotted  in  a  figure 
which  extends  well  over  the  sheet  of  squared  paper. 

When  a  curve  is  constructed  for  the  sole  purpose  of  reading 
off  intermediate  values,  a  large  square  should  represent  1,  5, 
10,  20,  50, 100  •  •  • ,  numbers  which  give  easy  readings. 

If  the  curve  is  made  simply  to  show  general  changes  or  to 
solve  a  problem,  the  unit  may  be  chosen  to  locate  the  points 
with  the  least  work. 

If  two  or  more  curves  are  constructed  on  the  same  axes,  they 
should  be  numbered  to  correspond  with  the  tables  or  data,  and 
they  can  be  more  readily  distinguished  if  a  different  kind  of 
line  is  used  for  each  curve,  for  example,  thick  and  thin  con- 
tinuous lines,  dotted  lines,  and  so  on.  When  convenient  the 
various  curves  may  be  drawn  in  different  colors ;  in  this  case 
the  table  of  values  should  be  written  in  the  same  color  that  is 
used  for  the  curve  which  represents  it. 


68 


APPLIED  MATHEMATICS 


EXERCISES 

1.  Construct  a  curve  from  the  record  of  temperature  given 
above  with  the  same  time  unit,  but  let  5  mm.  =  1°.  From  which 
curve  can  the  changes  be  read  most  easily  ? 

2.  Construct  several  temperature  curves  from  the  weather 
reports  in  the  daily  papers. 

3.  On  the  same  axes  construct  temperature  curves  for  a  day 
in  summer  and  a  day  in  winter. 

4.  Construct  on  the  same  axes  temperature  curves  for  several 
cities,  e.g.  Boston,  Chicago,  and  San  Francisco.- 

5.  Place  a  thermometer  outside  the  classroom  window  and 
take  readings  at  the  beginning  and  end  of  the  recitation  hour 
for  two  or  three  weeks.    Construct  the  curve. 

6.  Construct  several  curves  from  tables  found  in  newspapers, 
magazines.  The  Daily  Neivs  Almanac,  The  World  Almanac, 
Kent's  "  Mechanical  Engineers'  Pocket-Book,"  city,  state,  and 
government  reports,  price  lists,  and  so  on.  Try  to  find  reasons 
for  any  marked  peculiarities  in  the  curves. 

7.  Construct  curves  to  show  the  number  of  hours  of  daylight 
per  day  for  the  year.  (Let  a  heavy  horizontal  line  near  the 
center  represent  noon.  From  an  almanac  make  a  table  of  the 
time  of  sunrise  and  sunset  on  the  first  day  of  each  month; 
locate  the  points  and  draw  the  two  curves.)  On  the  same 
sheet  of  squared  paper  make  curves  for  different  latitudes, 
e.g.  Chicago  and  Dawson,  Alaska,  and  compare  the  amounts 
of  daylight. 

8.  A  price  list  of  the  Western  Electric  Company  gives  the 
following  price  of  bells.    Construct  the  curve. 


Size  of  gong  in  inches 
Price  in  dollars     .     . 


2i 
L68 


3 
1.74 


3i 
1.85 


5 
2.84 


3.20 


7 
4.56 


5.00 


10 
J.OO 


12 
10.00 


What  is  the  probable  price  of  a  9-in.  gong  ?  of  anll-in.gong  ? 


THE  USE  OF  SQUARED  PAPER 


69 


9.  The  water  in  a  glass  is  at  a  temperature  of  60°  F.  Heat 
is  applied  to  the  glass,  and  the  temperature,  T,  at  the  end  of 
t  minutes  is  as  follows  : 


Minutes 
Degrees 


0 
60 


10 
76 


15 
83.2 


20 
89.6 


25 
95.5 


30 
101 


35 
106 


40 
110 


Construct  the  curve.  What  temperature  would  you  expect 
at  the  end  of  7  min.  ?  of  32  min.  ? 

10.  A  boat  is  rowed  straight  across  a  river  and  soundings 
are  taken  at  various  distances  from  the  bank.  From  the  table 
draw  a  section  of  the  river  bed. 


Distance  from  bank  in  feet 
Depth  in  feet      .... 


20 


52 


11,  From  the  top  of  a  cliff  1500  ft.  high  a  bullet  was  shot 
horizontally  with  a  velocity  of  100  ft.  per  second.  Construct 
a  curve  to  show  its  path,  if  at  the  end  of  each  second  it  has 
fallen  the  following  number  of  feet : 


Number  of  seconds 
Distance  fallen 


3 
144 


4 

256 


5 
400 


576 


7 
784 


1024 


9 
1296 


10 
1600 


Take  the  x-axis  at  the  top  of  the  sheet.  On  the  cc-axis  let 
1  cm.  =  1  sec,  or  100  ft. ;  on  the  ?/-axis,  1  cm.  =  100  ft.  In  how 
many  seconds  will  the  bullet  reach  the  ground  if  it  is  level  ? 
How  far  from  the  foot  of  the  cliff  will  it  fall  ?  In  how  many 
seconds  will  it  fall  600  ft.  ?    How  far  will  it  fall  in  5\  sec.  ? 

II.  The  Graph  as  a  "Ready  Reckoner" 

47.  Straight-line  graphs.  In  the  following  exercises  the 
graph  is  a  straight  line.  Choose  convenient  units  and  let  the 
graph  extend  well  over  the  sheet  of  squared  paper. 


70 


APPLIED  MATHEMATICS 


EXERCISES 

1.  Construct  a  graph  to  change  inches  into  centimeters  and 
centimeters  into  inches,  given 
1  in.  =  2.5  cm. 


Construction. 

0  in.  =  0  cm. 
4  in.  =  10  cm. 

Locate  these  two  points,  O 
and  P,  and  draw  a  straight  line 
through  them.  Test  a  few 
points  on  the  graph  to  see  if  the 
results  are  approximately  cor- 
rect.   Thus  at  M  2  in.  =  5  cm. 


Q) 


p 


Inches 


Fig.  37 


2.  Construct   a    graph   to 

change  pints  to  liters,  given  that  11.  =  2.1  pt. 

3.  Construct  a  graph  to  lind  the  circumferences  of  circles  of 
diameter  from  0  to  18  in.,  given  that  the  circumference  equals 
TT  times  the  diameter. 

4.  Construct  a  graph  to  find  the  velocity  of  a  falling  body, 
given  that  the  velocity  at  any  second  equals  32  times  the 
number  of  seconds. 

5.  Construct  a  graph  to  change  miles  per  hour  to  feet  per 
second,  given  that  30  mi.  per  hour  equals  44  ft.  per  second. 

6.  Construct  a  graph  to  change  cents  to  marks,  given  that 
1  mark  equals  24  cents. 

7.  Construct  a  graph  to  change  cubic  inches  to  gallons,  given 
tliat  1  gal.  equals  231  cu.  in. 

8.  Construct  on  the  same  axes  graphs  to  find  the  simple  in- 
terest of  1 100  at  4  per  cent,  5  per  cent,  and  6  per  cent. 

9.  Construct  a  graph  to  find  the  number  of  amperes  in  a 
circuit  of  10  ohms  resistance  as  the  voltage  increases  from  10 
to  100  volts,  given  that  the  number  of  volts  divided  by  the 
number  of  ohms  equals  the  number  of  amperes. 


THE  USE  OF  SQUARED  PAPER  71 

10.  The  formula  for  the  number  of  revolutions  per  minute 

of  cutting  tools  in  lathes  is  n  =  -^-j-  >  where  n  =  revolutions 

per  minute,  s  =  the  speed  in  feet  per  minute,  and  d  =  the 
diameter  of  the  rotating  tool  in  inches.  Construct  a  graph  for 
a  tool  6  in.  in  diameter,  with  speeds  from  5  to  50  ft.  per 
minute. 

11.  The  resistance  /•  of  a  train  in  pounds  per  ton,  due  to 

s 
speed,  is  given  by  the  formula  r  =  3  +  -  •    Construct  a  graph 

for  speeds  from  5  to  60  mi.  per  hour. 

12.  The  pressure  of  the  atmosphere  in  pounds  per  square 
inch  for  readings  of  the  barometer  is  given  by  the  formula 
J)  =  .491  b,  where  p  =  the  pressure  in  pounds  per  square  inch, 
and  b  =  the  reading  of  the  barometer.  Construct  a  graph  for 
barometer  readings  from  28  in.  to  31  in.  Use  the  given  formula 
to  find  the  pressure  for  the  readings  28.75  in.,  29.50  in.,  and 
30  in.,  and  compare  with  the  pressures  read  from  the  graph. 

•  13.  Write  the  equations  which  express  the  relation  between 
the  two  quantities  in  each  of  the  preceding  exercises. 

Thus  in  Exercise  1  to  change  inches  to  centimeters  we  mul- 
tiply the  number  of  inches  by  2.5.  Therefore,  representing 
centimeters  by  c  and  inches  by  ^,  c  =  2.5  i  is  the  equation 
which  expresses  the  relation  between  centimeters  and  inches. 

48.  Equations  expressing  the  relation  between  two  quan- 
tities. In  the  first  list  of  exercises  the  curves  were  constructed 
from  tables  of  values  determined  by  observation  or  experiment. 
In  many  cases  there  is  no  known  relation  between  the  sets  of 
corresponding  numbers.  Thus  in  the  table  of  temperatures  the 
thermometer  was  read  at  intervals  of  one  hour,  and  we  do  not 
know  any  law  which  will  tell  what  the  reading  will  be.  But  in 
the  second  list  there  is  in  each  case  a  known  law  or  relation 
which  may  be  written  in  the  form  of  an  equation.  Thus 
1  in.  =  2.5  cm. ;  hence  the  number  of  centimeters  equals  the 


72  APPLIED  MATHEMATICS 

number  of  inches  multiplied  by  2.5,  or  c  =  2,5  i.  From  this 
equation  we  can  make  a  table  of  values,  and  from  the  table 
locate  points  and  construct  the  graph.  If  we  know  that  the 
graph  is  a  straight  line,  it  is  necessary  to  determine  only  two 
points  and  draw  a  straight  line  through  them. 

All  the  equations  in  this  exercise  are  of  the  first  degree 
and  all  the  graphs  are  straight  lines.  We  may  assume  that 
when  the  relation  between  two  quantities  is  expressed  by  an 
equation  of  the  first  degree  the  graph  is  a  straight  line 
(see  sect.  52  for  proof). 

49.  Curves.  When  the  equation  is  not  of  the  first  degree 
the  graph  will  be  a  curve  which  must  be  constructed  by  locat- 
ing a  number  of  points  sufficient  for  the  problem  in  hand. 

EXERCISES 

1.  Construct  a  curve  to  find  the  area  of  squares  whose  sides 
are  from  0  to  10  in.  (Let  1  cm.  horizontally  =  1  in.,  and  1  cm. 
vertically  =  10  sq.  in.)  If  a  =  the  area  and  s  =  a  side  of  th6 
square,  what  is  the  equation  that  connects  the  area  and  side  ? 
Find  from  the  equation  and  from  the  graph  the  area  of  a 
square  whose  side  is  (a)  3.5  in. ;  (b)  7.5  in. ;  (c)  9.25  in. 

2.  Construct  a  graph  to  find  the  surface  of  cubes  whose 
edges  are  from  0  to  10  in. 

3.  Construct  a  graph  to  find  the  area  of  circles  of  radii  from 
0  to  10  in.,  given  area  =  Trr^. 

4.  Construct  a  graph  to  find  the  volume  of  cubes  whose 
edges  are  from  0  to  10  in.  What  is  the  equation  connecting  v 
and  e? 

5.  Construct  a  graph  to  find  the  space  passed  over  by  a 
falling  body,  given  s  =  16t^,  t  =  number  of  seconds. 

6.  The  power  of  doing  work  possessed  by  a  body  in  motion 

(kinetic  energy)  is  given  by  ^  =  — —  >  where  w  —  the  weight 

^  g 


THE   USE  OF  SQUARED  PAPER  73 

in  pounds,  v  =  the  velocity  of  the  body  in  feet  per  second,  and 
g  =  32.  Construct  a  graph  to  show  the  kinetic  energy  of  a 
24-lb.  shot  as  its  velocity  changes  from  1600  to  600  ft.  per 
second. 

7.  The  volume  of  a  gas  diminishes  in  the  same  ratio  as  the 
pressure  on  it  is  increased,  or  jov  =  a  constant.  Given  pv  =  120, 
make  a  table  of  values  and  construct  a  curve  to  show  the 
volume  as  the  pressure  increases  from  1  lb.  to  60  lb.  per 
square  inch. 

8.  The  centrifugal  force  of  the  whole  rim  of  a  flywheel 

equals ?  where  iv  =  weight  of  the  rim  in  pounds,  r  =  mean 

.       ^''  .      . 

radius  of  the  rim  in  feet,  v  =  velocity  of  the  rim  in  feet  per 

second,  and  y  =  32.2.  Given  w  =  3220  lb.  and  r  =  5  ft.,  con- 
struct a  curve  for  velocities  from  10  to  100  ft.  per  second. 

9.  The  safe  load  in  tons,  uniformly  distributed,  on  horizontal 

11  bd^ 
yellow-pine  beams  is  to  =  —r^-r  >  where  b  =  breadth  of  beam 

lo  I 

in  inches,  d  =  depth  of  beam  in  inches,  and  I  =  distance  between 
the  supports  in  inches.  Construct  a  curve  to  show  the  safe  load 
on  yellow-pine  beams  4  in.  in  breadth,  12  ft.  between  supports, 
and  depths  from  8  to  18  in. 

10.  The  resistance  of  a  copper  wire  at  68°  F.  to  the  passage 

of  an  electric  current  is  given  by  i2  =      '      >  where  I  =  length 

of  wire  in  feet  and  d  =  diameter  of  wire  in  mils  (.001  in.). 
Construct  a  curve  for  the  resistance  of  1000  ft.  of  copper  wire 
of  diameter  from  5  to  100  mils. 

11.  The  volume  of  air  transmitted  in  cubic  feet  per  minute 
in  pipes  of  various  diameters  is  given  by  Q  =  .327  vd!^,  where 
V  =  velocity  of  flow  in  feet  per  second  and  d  =  diameter  of 
the  pipe  in  inches.  Construct  a  curve  to  show  the  volume  of 
air  transmitted  in  pipes  of  diameters  from  2  to  10  in.  with  a 
flow  of  12  ft.  per  second.  Without  further  computation  con- 
struct a  curve  for  a  velocity  of  24  ft.  per  second. 


74 


APPLIED  MATHEMATICS 


III.    The  Solution  of  Problems 

50.  In  a  graphical  solution  do  not  make  a  table  of  values 
unless  it  is  necessary. 

PROBLEMS 

1.  A  travels  6  mi.  per  hour  and  B  10  mi.  per  hour.  If  B 
starts  2  hr.  after  A,  when  and  where  will  they  meet  ? 

Solution.  Choose  units  and  axes  as  in  Fig.  38.  A  travels  24  mi.  in 
4  hr.  Locate  this  point  M,  and  draw  OA  through  the  points  0  and  M. 
B  starts  2  hr.  after  A ;  hence 
the  graph  of  his  journey  begins 
at  C.  He  travels  20  mi.  in  2  hr. 
Locate  this  point  iV  and  draw 
CB  through  the  points  C  and 
iV.  P,  the  intersection  of  OA 
and  CB,  shows  when  and  where 
they  meet,  —  5  hr.  after  A  starts 
and  30  mi.  from  the  starting 
point. 

The  figure  also  shows  how 
far  they  are  apart  at  any  time. 
Thus  at  the  end  of  3  hr.  they 
are  8  mi.  apart ;  this  number  of  miles  is  given  by  the  part  of  the  8-hr. 
line  included  between  the  lines  OA  and  CB. 

Solve  this  problem  and  some  of  the  others  in  this  list  algebraically 
and  compare  the  results  with  the  graphical  solution. 

2.  A  travels  7  mi.  per  hour  and  B  5  mi.  per  hour.  They 
start  at  the  same  time  and  travel  east,  A  from  a  town  M  and 
B  from  a  town  N  15  mi.  east  of  M.  When  and  where  will 
they  meet? 

3.  Two  trains  start  at  the  same  time  from  Chicago  and 
St.  Louis  respectively,  286  mi.  apart  ^  the  one  from  Chicago 
travels  50  mi.  per  hour  and  the  other  40  mi.  per  hour.  When 
and  where  will  they  meet  ? 

On  the  X-axis  let  a  large  square  =  20  mi.  Let  St.  Louis  be  at 
the  lower  left-hand  corner,  and  Chicago  14.3  squares  to  the  right. 


/ 

5 

30 

miles 

p 

^ 

N 

r<^ 

^ 

^ 

/ 

M 

to 

^ 

/ 

/ 

L 

0 

/" 

/ 

10 

z 

30  35   -OO 


Fig. 


THE  USE  OF  SQUARED  PAPER 


75 


Draw  the  line  to  represent  tlie  journey  of  the  St.  Louis  train  to  the 
right,  and  the  Chicago  train  to  the  left. 

4.  A  cyclist  starts  at  the  rate  of  300  yd.  per  minute,  and 
6  min.  later  another  cyclist  sets  off  after  him  at  the  rate  of 
500  yd.  per  minute.  When  and  where  will  they  meet  ?  When 
are  they  700  yd.  apart  ? 

5.  A,  traveling  20  mi.  per  day,  has  80  mi.  start  of  B,  who 
travels  25  mi.  per  day.    When  will  B  overtake  A  ? 

6.  A  invests  |500  at  6  per  cent  and  B  invests  flOOO  at 
5  per  cent.  In  how  many  years  will  A's  interest  differ  from 
B's  by  $300  ? 

Solution.  Choose  axes  and 
units  as  in  Fig.  39.  Interest  of 
$500  for  10  yr.  is  $300 ;  locate 
point  P,  and  draw  OA  through 
P  to  represent  A's  interest.  In 
a  similar  manner  draw  OB  to 
represent  B's  interest.  Three 
squares  vertically  represent 
$300.  Mark  off  three  squares 
on  the  edge  of  a  piece  of  paper 
and  with  it  find  on  what  verti- 
cal line  the  distance  between 
OA  and  OB  is  three  squares; 
result,  15  yr. 

7.  In  how  many  years  will  the  interest  on  $1500  at  5  per 
cent  be  $240  greater  than  the  interest  on  $1000  at  6  per  cent  ? 
When  will  it  be  $120  greater  ? 

8.  A  invests  $1000  at  5  per  cent  and  B  invests  $5000  at 
4  per  cent.  In  how  many  years  will  the  amount  of  A's  invest- 
ment equal  the  interest  of  B's  ? 

9.  A  invested  $2000  at  4 J  per  cent,  and  two  years  later  B 
invested  $2400  at  5  per  cent.  How  many  years  elapsed  before 
they  received  the  same  amount  of  interest  ?  When  was  the 
difference  of  the  interest  $120  ? 


Years 
Fig.  39 


76 


APPLIED  MATHEMATICS 


Y 

P^ 

/ 

\ 

// 

/ 

\ 

N 

^. 

A 

£^ 

/ 

\ 

^^ 

/ 

/ 

^ 

K 

\ 

J 

/ 

c 
1 

L 

a 

^ 

SH 

3       • 

^P^ 

I-. 

Fig.  40 


10.  A  man  walks  a  certain  distance  and  rides  back  in  8  hr. ; 
he  could  walk  both  ways  in  10  hr.  How  long  would  it  take 
him  to  ride  both  ways  ? 

Solution.  Let  OA  = 
10  hr.  (Fig.  40).  if  is  the 
mid-point  of  OA.  MP  is 
any  convenient  length. 
OP  A  represents  the  jour- 
ney when  the  man  walks 
both  ways,  and  OPB 
when  he  walks  and  rides 
back.  It  is  two  squares 
from  Ato  B;  take  C  two 

squares  from  O  and  join  CP.    Then  CPE  represents  the  journey  when 
he  rides  both  ways.   CB  =  6  hr.,  the  time  it  takes  him  to  ride  both  ways. 

11.  A  man  can  walk  to  Lincoln  Park  in  3^  hr.  If  he  walks 
to  the  park  and  rides  back  in  5;^  hr.,  how  long  would  it  take 
him  to  ride  both  ways  ? 

12.  A  man  walks  to  town  at  the  rate  of  4  mi.  per  hour  and 
rides  back  at  the  rate  of 
10  mi.  per  hour  after  re- 
maining in  town  1  hr. 
He  was  absent  8  hr. 
How  far  did  he  walk  ? 

Solution.  Choose  axes 
and  units  as  in  Fig.  41. 
OP  =  8  hr.  OA  is  the 
graph  of  the  walk  and  PB 
is  the  graph  of  the  ride.  If 
he  had  not  remained  in 
town,  the  distance  of  the 
point  of  intersection  of  the 
two  lines  from  the  x-axis 


Y 

35 

1 

/ 

30 

\ 

/ 

/ 

25 

\ 

/ 

/ 

20 

c 

X 

iH 

1,. 

y 

/ 

resf 

\ 

lO 

d> 

/ 

T 

5 

y 

■' 

\ 

/ 

N 

E_ 

3    -<*    e 
Hours 

Fig.  41 


would  give  the  distance  he  walked.  Since  he  remained  in  town  1  hr. 
we  find  where  the  horizontal  distance  from  OA  to  PB  equals  1  hr. 
This  is  CD  on  the  20-mi.  line ;  hence  the  man  walks  20  mi. 


THE  USE  OF  SQUARED  PAPER 


77 


13.  A  man  rides  to  a  city  at  the  rate  of  10  mi.  per  hour, 
remains  in  the  city  2  hr.,  and  returns  in  an  automobile  at  the 
rate  of  15  mi.  per  hour.  If  he  was  absent  10  hr.,  how  far  was 
it  to  the  city  ? 

14.  A  boy  starts  out  on  his  bicycle  at  the  rate  of  6  mi.  per 
hour.  His  wheel  breaks  down  and  he  walks  home  at  the  rate 
of  2^  mi.  per  hour.  How  far  did  he  ride  if  he  reached  home 
8^  hr.  after  starting  ? 

Construct  the  graphs  for  the  walk  and  ride,  as  in  Problem  10. 
The  intersection  of  the  lines  gives  the  distance. 

15.  A  man  rows  at  the  rate  of  6  mi.  per  hour  to  a  town  down 
a  river  and  2  mi.  per  hour  returning.  How  many  miles  distant 
was  the  town  if  he  was  absent  12  hr.  and  remained  in  town  6  hr.  ? 

16.  If  A  and  B  can  build 
a  sidewalk  in  6  and  4  da. 
respectively,  in  what  time 
can  they  build  it  working 
together  ? 

Solution.  Take  OX  in 
Fig.  42  any  convenient 
length,  and  let  OA  =  6  da. 
and  XB  =  4  da.  Draw  XA 
and  OB;  P  is  the  point  of 
intersection.  PM  =2.4  da., 
the  required  time. 

17.  A  can  do  some  work  in  30  da.  and  B  can  do  it  in  20  da. 
How  long  will  it  take  them  working  together  ? 

18.  If  A  can  do  some  work  in  12  hr.  that  he  and  B  can  do 
together  in  4  hr.,  in  what  time  can  B  do  it  ? 

As  in  Problem  14,  draw  XA  for  A's  work.  On  XA  take  P  4  units 
above  OX ;  draw  OP  and  produce  it  to  meet  XB  at  B.  XB  =  6  da., 
the  required  time. 

19.  Two  men  can  dig  a  ditch  in  8  da.  If  one  alone  can  dig 
it  in  40  da.,  how  long  will  it  take  the  other  man  to  dig  it  ? 


a'^ 


A 

X 

S, 

V 

X 

. 

4 
3 

2 

X 

s^ 

^^ 

^ 

^ 

^ 

^ 

^ 

N 

s 

^ 

, 

S 

N 

M 

K 

Fig.  42 


78 


APPLIED  MATHEMATICS 


20.  A  man  bought  100  lb.  of  brass  for  $13.60,  paying  for  the 
copper  in  it  16  cents  per  pound  and  for  the  zinc  10  cents  per 
pound.  How  many  pounds  of  each  metal  are  there  in  the  brass  ? 

Y 

Solution.  In  work- 
ing problems  take  the 
units  as  large  as  possi- 
ble ;  they  are  taken 
small  here  to  save  space. 
In  Fig.  43  OS  =  $13.60. 
OC,  OZ,  and  OB  are 
the  graphs  for  the  cop- 
per, zinc,  and  brass  re- 
spectively. Draw  BM 
parallel  to  OC,  intersect- 
ing OZ  at  P.  Draw 
PN±OX.  OiV^  =  401b., 


■    -* 

/ 

/c 

/ 

lO 

/ 

'A 

/ 

5° 

^5« 

/ 

^ 

y 

^ 

'^y 

^' 

>- 

^ 

^. 

y 

^ 

/■^ 

2 

.^ 

^ 

p 

^ 

0^ 

r 

N 

T 

30     40     60 
Pounds 


60      70    60     00    100  X 


Fig.  43 


the  number  of  pounds  of  zinc ;  and  NT  =  60  lb.,  the  number  of 
pounds  of  copper. 

Check.  .  40  -}-  60  =  100. 

40  X  .10  -I-  60  X  .16  =  13.60. 

Show  that  the  same  results  are  obtained  by  drawing  BM'  parallel 
to  OZ,  intersecting  OC  in  P'.  (A  geometrical  proof  of  the  construc- 
tion may  be  made  by  advanced  students.) 

21.  An  aluminum-zinc  alloy  weighing  300  lb.  was  sold  for 
$60,  the  cost  of  the  material.  If  the  aluminum  cost  25  cents 
per  pound  and  the  zinc  10  cents  per  pound,  how  many  pounds 
of  each  metal  were  in  the  alloy  ? 

22.  A  man  bought  100  A.  of  land  for  $3250.  If  part  of  it 
cost  him  $40  an  acre  and  part  of  it  $15  an  acre,  how  many  acres 
of  each  kind  were  there  ? 

23.  A  man  starts  off  rowing  at  the  rate  of  6  mi.  per  hour, 
and  half  an  hour  later  a  second  man  sets  out  after  him  at  the 
rate  of  8  mi.  per  hour,  (a)  When  is  the  first  man  overtaken  ? 
(b)  How  far  has  he  rowed  when  overtaken  ?  (c)  How  far  apart 
are  they  when  the  first  man  has  rowed  1  hr.  ? 


THE  USE  OF  SQUARED  PAPER 


79 


24.  The  distance  from  Chicago  to  Milwaukee  is  85  mi.  An 
automobile  leaves  Chicago  at  1.00  p.m.  at  the  rate  of  15  mi. 
per  hour  and  another  leaves  Milwaukee  at  1.30  p.m.  at  the 
rate  of  18  mi.  per  hour.    When  and  where  will  they  meet  ? 

25.  A  man  walked  to  the  top  of  a  mountain  at  the  rate  of 
2^  mi.  per  hour,  and  down  the  same  way  at  the  rate  of  3^  mi, 
per  hour.    If  he  was  out  5  hr.,  how  far  did  he  walk  ? 

26.  From  the  same  place  on  a  circular  mile  track  two  boys, 
A  and  B,  start  at  the  same  moment  to  walk  in  the  same  direc- 
tion, A  4  mi.  per  hour  and  B  3  mi.  per  hour.  How  often  and 
at  what  times  will  they  meet  if  they  walk  1^  hr.  ? 

27.  If  the  two  boys  in  Problem  26  walk  in  opposite  directions 
around  the  track,  how  often  and  at  what  times  will  they  meet  ? 

28.  A  with  an  old  automobile  travels  15  mi.  an  hour,  and 
stops  5  min.  at  the  end  of  each  hour  to  make  repairs.  B  on  a 
new  car  travels  25  mi.  per  hour.  If  B  starts  3  hr.  after  A, 
when  and  where  will  he  overtake  A? 


IV.   The  GrRAPHicAL  Kepresentation  and  Solution 
OF  Equations 

51.  Equations  of  the  first  degree.  We  have  graphed  equa- 
tions which  arose  in  concrete  problems,  and  we  will  now  apply 
the  same  methods  to  abstract  equations  containing  the  two 
unknowns  aj^nd  y. 

Exercise.    Construct  the  graph  of  a;  -f-  y  =  5. 

Transposing,  y  —  h  —  x.  , 

By  giving  values  to  x  we  have  the  following  table : 


X 

8 

7 

6 

5 

4 

3 

2 

1 

0 

-1 

-2 

-3 

V 

-3 

-2 

-1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

For  the  first  time  in  our  graphical  work  we  have  to  deal  with 
negative  numbers.    This  will  cause  no  trouble,  however,  for  we 


80 


APPLIED  MATHEMATICS 


^ 

k° 

\ 

^ 

k 

2 

1 

k 

?«'-« 

-J. 

-? 

o 

2 

■a, 

k« 

B 

-2 

1 

V 

\ 

-e 

-e 

Y 

Fig.  44 


will  simply  count  off  the  positive  values  of  x  to  the  right  of 
the  origin,  and  the  negative  values  to  the  left.  For  positive 
values  of  y  count  up  from  the 
a;-axis,  and  for  negative  values 
count  down. 

Taking  heavy  horizontal  and 
vertical  lines  near  the  center  of 
the  page  for  the  a;-axis  and  ?/-axis 
respectively,  locate  the  points 
from  the  table  and  draw  a  line 
through  them.  The  axes  should 
always  be  lettered  as  in  Fig.  44, 
and  the  units  indicated  on  the 
axes  or  on  the  sides  of  the 
diagram. 

It  is  not  worth  while  to  plot  many  equations  of  the  first 
degree  by  locating  points,  since  it  will  be  proved  in  the  next 
paragraph  that  such  a  graph  is  always  a  straight  line.  Hence 
in  plotting  equations  of  the  first  degree  it  is  necessary  to  locate 
only  two  points.  These  points  should  be  some  distance  apart 
in  order  that  the  graph  may  be  fairly  accurate. 

52.  Theorem.  The  graph  of  an  equation  of  the  first  degree 
is  a  straight  line. 

Proof.    Any  equation  of  the  first  degree  can  be  reduced  to  the  form 

y  =  mx  +  b  (1) 

by  transposing,  uniting,  and  divid- 
ing by  the  coefficient  of  y.  Let  P  be 
a  point  on  the  graph  oi  y  =  mx  +  h. 
Draw  PM  J.  OX.  Then,  for  the 
point  P,  OM  =  X  and  PM  =  y.  In 
equation  (1)  put  x  =  0  ;  then  y  =  b, 
that  is,  the  graph  of  (1)  cuts  the 
y-axis  at  the  point  (0,  b).  Let  OA  =  b. 
Through  P  and  A  draw  the  straight 
line  AC.   Through  A  draw  AF  parallel  to  OX,  cutting  PM  at  N. 


THE  USE  OF  SQUARED  PAPER 


81 


From  (1), 

From  the  figure, 
and 

Therefore 


y 


y-h  =  PN, 

x  =  AN. 

PN  _y-h 

AN~     X 


Why? 
Why? 


That  is,  for  any  point  P  on  the  graph  of  y  =  mx  +  b  the  ratio 
PN/AN  is  constant,  since  m  is  some  fixed  number.  Hence,  by  the 
properties  of  similar  triangles  (what  are  they  ?),  any  point  whose  x 
and  y  satisfy  equation  (1)  lies  on  the  straight  line  A  C. 


EXERCISES 

Plot  the  following  equations  : 

1.  X  -\-  y  =  Q.  Z.  X  -\-  y  =  —  Q. 

2.  X  —  y  =  6.  A.   —  X  -\-  y  =  6. 

53.  Equations  of  degree  higher  than 
the  first.  The  graph  of  an  equation  of 
degree  higher  than  the  first  is  a  curve, 
which  can  be  drawn  with  sufficient  accu- 
racy by  locating  a  number  of  points. 

Exercise.    Plot  y  =  x^  —  6x  +  5. 

If  we  wish  to  take  the  side  of  a  large 
square  =  1  on  both  axes,  it  is  necessary 
to  begin  the  table  of  values  with  some 
value  of  X  that  will  bring  the  point  on 
the  paper.  If  we  start  with  x  =  S,  then 
y  =  21,  and  the  point  (8,  21)  is  off  the 
paper ;  hence  we  begin  with  x  =  7. 


5.  2x  +  3y=6. 

6.  5x-7y  =  35. 


Y 

lO 

e 

6 

• 

1 

1 

i 

/ 

-^ 

-a 

oV_ 

2 

J 

©    X 

^' 

-?\ 

\ 

-A 

U 

Y 

-6 

Fig.  46 


7 
12 


-1 
12 


82 


APPLIED  MATHEMATICS 


Usually  it  is  necessary  to  locate  points  close  together  to 
determine  the  true  shape  of  the  curve  at  some  particular  point. 
Thus  from  the  given  equation  : 


3.5 
3.75 


3.2 
3.96 


3.1 
3.99 


2.9 
3.99 


2.8 
3.96 


2.5 

3.75 


These  additional  points  show  that  the  curve  is  rounded  at 
(3,  —  4).    This  point  is  called  the  turning  point  of  the  curve. 

54.  The  purpose  of  graphical  representation.  From  this 
curve  we  may  learn  two  things  :  (1)  the  x  of  the  points  where 
it  intersects  the  ic-axis,  1  and  5,  are  the  roots  of  the  equation 
a;^— 6ic  +  5  =  0;  (2)  the  y  of  the  turning  point,  —  4,  gives  the 
least  value  of  the  expression  cc*^  —  6  a;  +  5  (see  Chapter  VIII). 

EXERCISES 

Plot  these  equations.  In  the  first  four  find  the  least  value  of 
the  expression  and  the  roots  of  the  equation  when  ?/  =  0 : 

1.  y  =  x^  —  ^x  —  b.  5.  a;^  -h  2/^  =  25  (circle). 

2.  ?/  =  a;^  —  6  a;  +  9.  6.  3/^  =  8  a;  (parabola). 

Z.  y  =  x'-x-Q.  7.  9 a;*^  +  25 2/^  =  225  (ellipse). 

4.  y  =  a;^  -(-  a;  —  2.  8.  4  a;^  —  9  7/^  =  36  (hyperbola). 

55.  A  short  method  of  computing  the  table  of  values  for 
equations  of  degree  higher  than  the  second.  This  method  can 
be  used  also  in  checking  the  roots  of  equations. 

Exercise  1.    Plot  ?/ =  a;^  -  5  a;^  -  2  a;  +  24. 

Let  X  =  6.  x^  =  xx^  =  6  x^. 

...  a;8  -  5  x2  -  2  X  +  24  =  6  x2  -  5  x2  -  2  X  +  24 
=  x2  -  2  X  +  24. 
x'  =  XX  =  6  X. 
.-.  x2  -  2  X  +  24  =  6  X  -  2  X  +  24  =  4  X  +  24. 
4  X  =  4  X  6. 
.-.  4  X  +  24  =  24  +  24  =  48. 

.-.  y  =  48  when  x  =  6. 


THE  USE  OP^  SQUARED  PAPER 


83 


The  coefficients  only  need  be  written  and  the  work  can  be 
put  in  the  following  form  : 

1-5-2  + 24  [6 
6  +  6  +  24 
1  +  4  +  48 

After  the  coefficients  are  written  we  multiply  the  first  one 
at  the  left  by  6  and  add  the  product  to  the  second,  obtaining  1. 
This  sum  is  multiplied  by  6  and  added  to  the  third  coefficient, 
and  so  on. 

If  any  power  of  x  is  lacking,  write  0  for  the  coefficient  of 
the  missing  term.  Thus,  if  ?/  =  a;*  +  3  ic^  +  2  x  +  5,  write  the 
coefficients  1  +  0  +  3  +  2  +  5. 

Table  of  Values  fou  ?/  =  a;^  —  5  x^  —  2a;  +  24 


X 

6 

5 

4 

^ 

3 

2 

1 

0 

-1 

-2 

-3 

y 

48 

14 

0 

-If 

0 

8 

18 

24 

20 

0 

-42 

Locate  axes  and  choose  conven- 
ient units,  as  in  Fig.  47.  Since  2/  =  0 
for  X  =  4:  and  a;  =  3,  it  is  necessary 
to  locate  one  or  more  points  between 
X  =  4:  and  a;  =  3  to  get  the  curve 
fairly  accurate.  The  roots  of  the 
equation  x^  —  5  a;^  —  2  x  +  24  =  0  are 
seen  to  be  —  2,  3,  and  4. 

Exercise  2.  Plot  ^  =  a-^  -  6  a;^  —  2. 

Exercise  3.  Plot 
2/  =  a;*  +  a;"  -  13  a;2  -  a;  +  12. 


55- 

Y 

/ 

z5 

i\ 

V 

2o 

/ 

\ 

IS"  ■ 

\ 

]0 

{ 

/ 

5- 

V 

/ 

X 

o 

V 

-s 

=3^ 

-a 

o 

z 

^ 

6   .. 

Fig.  47 


Table  of  Values 


X 

4 

3.5 

3 

2 

1 

0 

-1 

-2 

-3 

-4 

-4.6 

y 

120 

44.2 

0 

-18 

0 

12 

0 

-30 

-48 

0 

72 

84 


APPLIED  MATHEMATICS 


Y 

/ 

4ol 

/ 

30  1 

/ 

ZO    \ 

,^ 

>. 

/ 

10 
< 

/ 

\ 

[ 

X 

0 

'     o 

\ 

j 

-» 

V 

y 

•20 

1 

-30 

\ 

/ 

-40 

-4 

\^ 

V. 

,..,  -( 

9Y'    , 

....? 

..  ^ 

_.  ■« 

Find  the  table  of  values  by  the  short  method.  The  choice  of 
units  in  Fig.  48  makes  the  curve  of  good  form  for  a  study  of 
its  properties.  The  roots 
of  the  equation  x*  +  ^* 
-13a;2-a;  +  12  =  0  are 
seen  to  be  —  4,  —  1, 1,  and 
3.  How  can  the  position 
of  the  three  turning  points 
be  found  ? 

56.  Helpful  principles 
in  plotting  curves.    For 

equations  in  the  form  y 
equal  an  expression  con- 
taining cc,  with  no  root 
signs  and  no  term  in  the 

"Fir*    48 

denominator  containing  cc, 

the  following  principles  are  useful  in  plotting  the  curves : 

1.  The  number  of  turning  points  cannot  be  greater  than  the 
degree  of  the  equation  less  one.  Thus  an  equation  of  the 
fourth  degree  cannot  have  more  than  three  turning  points. 

2.  A  line  parallel  to  the  y-axis  can  cut  the  curve  only  once. 

3.  If  the  equation  is  of  odd  degree,  the  ends  of  the  curve 
are  on  the  opposite  sides  of  the  ic-axis. 

4.  If  the  equation  is  of  even  degree,  both  ends  of  the  curve 
are  on  the  same  side  of  the  ic-axis. 

5.  The  number  of  times  the  curve  cuts  the  ic-axis  cannot  be 
greater  than  the  degree  of  the  equation. 

EXERCISES 

Construct  curves  to  represent  the  following  equations  : 

1.  y  =  x^  +  2  x^  —  X  —  2.  A.  y  =  x^  —  Ax^. 

2.  y  =  x^  +  x''-x-l.  5.  2/  =  a;'  -  10a;2  -f  8. 

3.  2/ =  a;"  +  3a:' -6a; -8.  6.  y  =  x*  -  4a;2  +  4a;  -  4. 


THE  USE  OF  SQUARED  PAPER 


85 


57.  Solution  of  simultaneous  equations.    Equations  like 


X  -\-  y  =  8 
2x  +  37/  =  18 


and 


x^-i-i/  =  25 
3x  +  Ai/  =  25 


can  be  solved  by  plotting  the  curves  on  the  same  axes  and  not- 
ing where  they  intersect.  The  x  and  the  y  of  each  point  of  in- 
tersection gives  a  pair  of  values  which  satisfies  each  equation. 
The  graphical  solution  shows  clearly  how  many  pairs  of  values 
there  are,  and  why  a  certain  value  of  x  must  be  taken  with  a 
certain  value  of  y.  In  many  cases,  however,  the  algebraic  solu- 
tion can  be  made  more  quickly.  But  squared  paper  is  of  real 
service  in  solving  equations  of  degree  higher  than  the  second 
containing  one  unknown. 

58.  Solution  of  equations  of  any  degree ;  real  roots.  The 
principle  involved  in  graphical  solution  is  readily  seen  by  look- 
ing at  the  curves  already  plotted.  Suppose  we  wish  to  solve 
the  equation  jc^  —  6a;-|-5  =  0;  that  is,  we  want  to  find  values 
of  X  which  make  the  expression  x^  —  6x  -\-  5  zero.  Put  y  = 
x^  —  6x  -{-  5  and  we  obtain  the  curve  in  Fig.  46.  At  the  point 
where  the  curve  cuts  the  x-axis  y  is  0.  Since  the  curve  cuts 
the  ic-axis  at  a;  =  1  and  a;  =  5,  the  solutions  ofa;^  —  6x-|-5  =  0 
are  1  and  5.  Look  over  the  curves  you  have  plotted  and  de- 
termine the  solutions  when  possible.  If  the  roots  of  an  equa- 
tion are  small  whole  numbers,  they  can  easily  be  found  by 
factoring  the  given  expression.  If  the  given  expression  cannot 
be  factored,  the  roots  can  be  found  to  as  many  decimal  places 
as  are  needed  by  graphical  methods. 

Exercise.    Solve  a;«  -  5  a;"  -  2  x  +  20  =  0. 

Put  y  =  x^  —  5  x^  —  2  X  +  20  aud  compute  the  following  table  of 
values : 


X 

5 

4.5 

4 

3.5 

3 

2.5 

2 

1 

0 

-1 

-1.5 

-2 

y 

10 

.875 

-4 

-  5.375 

-4 

-  .625 

4 

14 

20 

16 

,    8.375 

-4 

86 


APPLIED  MATHEMATICS 


Time  is  saved  by  plotting  the  curve  rather  accurately  where 
it  cuts  the  X-axis. 

Fig.  49  shows  that  the  roots  of  the  equation  lie  between  4 
and  5,  2  and  3,  and  —  1  and  —  2.  We  will  find  the  first  root 
to  two  decimal  places.  Since  the  curve  seems  to  cut  the  ic-axig 
between  x  =  4.4  and  x  —  4.5,  we  substitute  these  two  values  in 


Y 

r 

\ 

/ 

'     16 

> 

\ 

8 

/ 

IZ 

\ 

A 

i^ 

y' 

/ 

8 

\ 

1 

o 

/ 

X 

4 

\ 

X 

-4 

■*3 

4.4 

r 

-3 

-z 

-1 

O 

1 

z 

\ 

A 

r 

p 

-4 

\>^ 

-a 

Fig,  49 


the  equation,  obtaining  for  x  =  4.4,  y  =—  .416 ;  and  for  x  =  4.5, 
y  =  .875.  The  change  in  sign  shows  that  the  curve  does  cut 
the  ic-axis  between  these  two  points,  and  the  root  to  two  figures 
is  4.4. 

The  next  thing  is  to  draw  the  part  of  the  curve  between  x  =  4.4 
and  X  =  4.5  to  a  larger  scale,  as  in  Fig.  49.  The  two  points  P 
and  P'  may  be  joined  by  a  straight  line  which,  in  general,  will 
lie  close  to  the  curve.  The  curve  seems  to  cross  the  cc-axis  be- 
tween X  =  4.43  and  x  =  4.44.  For  x  =  4.43,  y  =  —  .0462 ;  and 
for  x  =  4.44,  2/  =  .0803.  The  change  of  sign  shows  that  the 
curve  does  cross  the  a;-axis  between  these  two  values  of  x. 
Hence  the  root  to  two  decimal  places  is  4.43.  In  a  similar 
manner  the  root  could  be  found  to  any  desired  number  of 
decimal  places. 

Find  tke  other  two  roots  to  two  decimal  places. 


THE  USE  OF  SQUARED  PAPER 


87 


PROBLEMS 

Find  the  roots  of  these  equations  to  three  decimal  places : 

1.  a;»-3a;2-2a;4-5  =  0  (root  between  1  and  2). 

2.  cc*  —  4a;^  —  6a;-f-8  =  0  (root  between  4  and  6). 

3.  03^  +  2  ic^  —  4  X  —  43  =  0  (positive  roots), 

4.  cc*  —  12  cc  +  7  =  0  (positive  roots). 

5.  cc^  —  5x^  +  803  —  1  =  0  (root  between  0  and  1). 

6.  x^  +  2x^-Sx-9  =  0  (root  between  1  and  2). 

7.  ic^  —  7a;  +  7  =  0  (root  between  —  3  and  —  4). 

8.  a;'  -  2a;'-'  -  a;  +  1  =  0  (3  roots). 

9.  a;8  -  3x  +  1  =  0  (3  roots). 


V.    Determination  of  Laws  from  Data  obtained  by 
Observation  or  Experiment 

59.  Exercise.    Find  the  law  of  a  helical  spring. 

In  the  physics  laboratory  a  helical  spring  was  loaded  with 

weights  of  100  g.,  200  g.,  •••,  and  the  elongation  for  each  load 

was  recorded  in  the  following  table  : 


X  (grams)   .     . 
y  (centimeters) 


100 


200 
3 


300 
6,4 


400 
10.4 


500 
14.5 


600 
18,6 


700 
22.6 


800 
26,8 


900 
30,9 


Plot  these  points  care- 
fully, choosing  the  units 
to  get  as  large  a  figure  as 
possible.  Stretch  a  fine 
thread  along  the  points 
and  it  will  be  found  that 
it  can  be  placed  so  that 
most  of  the  points  will  lie 
close  to  it  or  on  it,  and 
that  they  will  be  rather 


35 

3o 

y 

Y 

A 

Y 

A 

Y 

A 

Y 

lO 

/ 

y 

5 

o      i 

\^ 

200 

Qroms 

aseJ 

gfiSL. 

aasL. 

FiG,  50 


88  .     APPLIED  MATHEMATICS 

evenly  distributed  above  and  below.  Hence  it  is  evident  that  an 
equation  of  the  first  degree  connects  the  grams  and  centimeters. 
In  this  statement  the  first  two  loads  are  omitted,  and  no  load 
greater  than  900  g.  is  considered,  since  at  that  load  the  spring 
showed  signs  of  breaking.  Draw  a  straight  line  in  the  position 
of  the  thread. 

Let  us  suppose  that  the  law  or  equation  is  in  the  form 

y  =  mx  +  h.  (1) 

The  values  of  m  and  b  must  be  found  that  will  best  fit  the  data. 
Take  two  points  which  lie  close  to  the  straight  line  and  some 
distance  apart,  and  substitute  the  x  and  y  of  these  points  in  (1). 
Taking  the  fourth  and  ninth  points,  we  have 

10.4  =  400  m  +  b.  (2) 

30.9  =  900  m  +  b.  (3) 

(3)  -  (2),  20.5  =  500  m.  (4) 

m  =  .061.  (5) 

Substituting  (5)  in  (2),      b  =  -6.  (6) 

Therefore  y  =  .041  x  —  6  is  the  required  equation  or  law. 

Check.   Substitute  the  x  and  y  of  sixth  point. 

18.6  =  600  X  .041  -  6 
=  18.6. 

Substitute  the  x  and  y  of  the  seventh  point,  we  obtain  22.6  =  22.7. 

60.  Straight-line  laws.  When  the  results  of  experimental 
work  are  plotted  it  frequently  happens  that  the  points  lie  nearly 
in  a  straight  line.  In  such  cases  it  is  not  difficult  to  find  the 
law  or  equation  by  the  method  used  in  the  preceding  exercise. 
Since  there  are  always  errors  in  experimental  work  the  points 
will  not,  of  course,  lie  exactly  in  a  straight  line.  If  some  of 
the  points  lie  at  a  rather  large  distance  from  the  straight  line 
through  several  of  them,  it  may  be  that  the  equation  is  not 
of  the  first  degree.  In  the  following  exercises  the  graphs  are 
straight  lines. 


THE  USE  OF  SQUARED  PAPER 


89 


EXERCISES 

1.  Make  a  helical  spring  by  coiling  a  wire  around  a  small 
cylinder.  Arrange  the  spring  to  carry  a  load ;  take  readings  of 
the  elongation  for  several  loads  and  find  the  law  of  the  spring. 

2.  Put  a  Fahrenheit  and  a  Centigrade  thermometer  in  a 
dish  of  water  and  take  the  reading  of  each.  Vary  the  tempera- 
ture of  the  water  by  adding  hot  water  or  ice  and  take  sev- 
eral readings.  Find  the  law  connecting  the  readings  of  the 
two  thermometers. 

3.  Load  a  thin  strip  of  pine  supported  at  points  two  feet  apart 
and  note  the  deflection.  Vary  the  load  and  find  that  for  loads 
under  a  certain  weight  the  deflection  is  proportional  to  the  load. 
For  what  weight  does  the  law  begin  to  fail  ? 

4.  Find  the  laws  of  the  following  helical  springs  : 


X  (ounces) 

4 

5 

6 

7 

8 

9 

10 

1 

y  (Inches) 

5.2 

5.5 

5.8 

6.1 

6.4 

6.7 

7 

2 

y  (inches) 

13.2 

14.0 

14.8 

15.6 

16.4 

17.2 

18 

3 

y  (inches) 

3.8 

5.0 

6.2 

7.4 

8.6 

9.8 

10 

5.  I  is  the  latent  heat  of  steam  in  British  thermal  units 
(B.  t.  u.)  at  f  F.    Find  an  equation  giving  I  in  terms  of  t. 


170.1 
995.2 


193.2 
979.0 


212.0 
965.7 


240.0 
945.8 


254.0 
935.9 


6.  V  is  the  volume  of  a  certain  gas  in  cubic  centimeters  at 
the  temperature  f  C.  If  the  pressure  is  constant,  find  the  law 
connecting  F  and  t. 


i 
V 


27 
110 


33 

112 


40 
115 


120 


68 
125 


90 


APPLIED  MATHEMATICS 


7.  A  steel  bar  107  cm.  long  was  supported  at  the  ends  and 
loaded  at  the  center  with  the  following  results.  Find  the  equa- 
tion connecting  the  load  and  deflection. 


Grams .     . 
Deflection 


500 
1.18 


1000 
2.35 


2000 
4.72 


3000 

7.15 


4000 
9.42 


8.  In  an  arc-light  dynamo  test  the  voltage  for  the  revolutions 
per  minute  was  recorded.  Find  the  laws  connecting  the  volts 
and  revolutions  per  minute. 


Revolutions  per  minute 
Volts 


•200 
165 


300 
253 


400 
337 


500 
421 


600 
507 


700 
590 


9.  P  is  the  pull  in  pounds  required  to  lift  a  weight  W 
by  means  of  a  differential  pulley.  Find  the  law  connecting 
P  and   W. 


W 
P 


50 
8.0 


100 
13.4 


150 
19.0 


200 
24.4 


250 
30.1 


300 
■35.6 


10.  When  the  weight  W  was  lifted  by  a  laboratory  crane 
the  force  applied  to  the  handle  was  P  pounds.  Find  the  law 
connecting  P  and  W. 


W 
P 


50 
7.4 


100 
8.3 


150 
9.5 


200 
10.3 


250 
11.6 


300 
12.4 


350 
13.6 


400 
14.6 


CHAPTER   VIII 

FUNCTIONALITY;  MAXIMUM  AND  MINIMUM  VALUES 

61.  Number  scale.  Real  numbers  are  represented  graphi- 
cally by  a  straight-line  scale.  Zero  is  the  dividing  point  between 
the  positive  and  the  negative  field,  and  may  be  considered  either 
positive  or  negative. 

In -going  down  the  negative  scale  further  and  further  from  zero 
the  numbers  are  getting  smaller ;  that  is,  —  10  is  less  than  —  3. 
The  actual  magnitude  of  a  number,  without  regard  to  its  sign 
or  quality  or  position  in  the  scale,  is  called  its  absolute  value. 

-Qj  -A       -3       -2      -I  p      -t-i     .+2        43       t4  -too 

Fig.  51 

Beginning  at  the  extreme  left  and  passing  constantly  to 
the  right,  numbers  may  be  said  to  increase  continuo^ishj  from 
—  00  through  0  to  -)-  co.  Beginning  at  the  extreme  right  and 
passing  constantly  to  the  left,  numbers  may  be  said  to  decrease 
continuously  from  +  oo  through  0  to  —  co.  Beginning  at  any 
point  and  passing  to  the  right  gives  increasing  numbers,  while 
passing  to  the  left  gives  decreasing  numbers. 

62.  Variables.  A  variable  is  a  number  which  changes  and 
passes  through  a  series  of  successive  values.  It  may  pass 
through  the  whole  scale  of  values  from  —  oo  to  -f  oo,  or  it 
may  pass  through  a  certain  portion  of  the  scale  only.  If  the 
variable  is  confined  to  a  portion  of  the  number  system,  as 
from  the  position  —  15  in  the  scale  to  the  position  -|-  6,  it  is 
said  to  have  the  interval  —  15  to  +  6. 

A  number  is  said  to  vary  continuotisly  in  a  given  interval, 
a  to  b,  if  it  starts  with  the  value  a  and  increases  (or  decreases) 

91 


92 


APPLIED  MATHEMATICS 


to  the  value  b  in  such  a  way  as  to  assume  all  values  between 
a  and  b  (integral,  fractional,  and  irrational)  in  the  order  of 
their  magnitude. 

63.  Inequality  of  numbers.  One  number  is  greater  than  a 
second  if  a  positive  number  must  be  added  to  the  second  to 
produce  the  first.  Thus  —  3  is  greater  than  —  8,  since  +  5  must 
be  added  to  —  8  to  obtain  —  3. 

One  number  is  less  than  a  second  if  a  positive  number  must 
be  subtracted  from  the  second  to  obtain  the  first.  Thus  —  17 
is  less  than  —  12,  since  +  5  must  be  subtracted  from  —  12  to 
obtain  —  17. 

The  relation  of  inequality  is  usually  expressed  by  a  symbol. 
Thus  -  3  >  -  8,  10  >  4,  -  17  <  -  12,  2  <  7. 

64.  Function  of  a  variable.  The  value  of  an  expression  in- 
volving a  variable  depends  upon  the  value  of  the  variable. 
The  expression  is  called  a  function  of  the  variable.  Thus 
x^  —  1  is  a  function  of  x  (written  f(x)  =  a;^  —  1,  and  read 
"  function  of  x  equals  x^  —  1 "),  for  when  x  has  the  values 
—  2,-1,  0,  4-1,  +2  respectively,  x^  —  1  has  the  values  3, 
0,  -  1,  0,  3. 

The  variable  to  which  we  may  give  values  at  will  is  called 
the  independent  variable  ;  but  the  expression  or  variable  which 
depends  upon  it  for  its  value  is  called  the  dependent  variable, 
or  function.  The  volume  of  a  cube  is  a  function  of  the  edge, 
V  =  f{e)  ==  e^.  The  area  of  a  circle  is  a  function  of  the  radius, 
a  =  f(r)  =  7rr^.  The  distance  through  which  a  body  falls  is  a 
function  of  the  time,  s  =f(t)=  \  gf.  Name  the  independent 
and  dependent  variables  in  the  preceding  illustrations. 

Exercise.  Plot  the  graph  of  the  function  2  jc'  —  3  a;^  —  12  a;  +  4. 

Give  X  integral  values  from  —  3  to  4  and  obtain  the  following  table : 


2x3-3x2-12x  +  4 


-3 
-41 


2 
-16 


4 
36 


FUNCTIONALITY 


93 


You  have  been  constructing  curves  by  locating  points  from 
a  table  and  drawing  a  smooth  curve  through  them ;  you  should 
now  see  that  this  method  of  plotting  a  function  is  based  on  the 
assumption  that  the  given  expression  is  a  continuous  function  of 
X.  In  this  case  a  small  change  in  x  makes  a  small  change  in  the 
given  function ;  hence  if  all  values  of  x  were  taken,  there  would 
be  a  continuous  succession  of  points  forming  a  smooth  curve. 

In  Fig.  52  imagine  a  perpendicular  to  the  a-axis  drawn  to 
the  curve  from  x  =—  3.  The  length  of  this  perpendicular  is 
the  value  of  the  function 
for  X  =  —  3.  Now  imagine 
the  perpendicular  to  move 
to  the  right  to  cc  =  +  4,  and 
you  have  a  mental  picture 
of  the  function  varying  con- 
tinuously in  value  from  —  41 
to  +  11,  then  to  —  16,  and 
finally  to  +  36. 

For  certain  intervals  of 
values  of  x  the  function  is 
greater  than  zero,  and  for 
certain  intervals  it  is  less 
than  zero.  For  certain  defi- 
nite values  of  x  the  function 

has  the  value  zero.  The  value  of  the  function  is  greater  than 
zero  in  the  intervals  from  x  =  —  2  tox  =  A  (about),  and  from 
x  =  2.9  (about)  to  cc  =  -f  oo.  The  function  is  less  than  zero  from 
a;=— 00  tocc=—  2,  and  from  x  =  A  (about)  to  a;  =  2.9  (about). 
The  function  has  the  value  zero  for  x  =—  2  and x  =  A  (about). 

65.  Maximum  and  minimum  values.  As  x  increases  from 
—  3  to  —  l,2x^  —  3x^  —  12x  +  A:  increases  from  —  41  to  +  11. 
As  X  increases  from  —  1  to  +  2,  the .  function  decreases  from 
-I-  11  to  —  16.  As  X  increases  from  +  2  to  +4,  the  function 
increases  from  —  16  to  +36.  We  observe  that  as  the  variable 
X  increases  continuously,  the  value  of  the  function  may  either 


30 

Y 

—J^ 

r»v~ 

0 
-10 

x' 

f 

N 

^, 

\ 

X 

/ 

6 

N 

I 

/ 

\ 

J 

-20 
-30 
./|0 

/ 

\ 

/ 

y' 

•♦2 

\     -, 

s      - 

\       < 

> 

2 

\     : 

»     ^ 

\ 

Fig.  52 


94  APPLIED  MATHEMATICS 

increase  or  decrease.  At  any  point  where  the  function  stops 
increasing  and  begins  to  decrease,  it  is  said  to  have  a  maximum 
value  or  to  be  a  Tnaximum.  In  this  case  it  occui's  when  x  =—  1, 
or  when  the  function  has  the  value  +  11. 

When  the  function  stops  decreasing  and  begins  to  increase, 
it  is  said  to  have  a  minimum  value  or  to  be  a  7ninlmum.  Here 
it  occurs  when  x  =  2,  or  when  the  function  has  the  value  —  16. 

In  other  words,  a  function  is  a  maximum  when  its  value  is 
greater  than  the  values  immediately  preceding  and  following. 
In  the  same  way  a  function  is  a  minimum  when  its  value  is 
less  than  the  values  immediately  preceding  and  following. 
The  point  on  the  curve  at  which  there  is  a  maximum  or 
minimum  value  of  the  function  is  called  a  Uirning  jjoint. 

66.  To  investigate  functional  variation  and  get  an  idea  of 
regional  increase  and  decrease,  and  maximum  and  minimum 
values.  Plot  enough  points  to  give  the  shape  of  the  curve.  The 
regions  of  increase  and  decrease  are  then  readily  noted.  To  check 
an  apparent  maximum  or  m,inimum  vahie  of  the  function,  cal- 
culate values  of  the  function  for  points  close  together  in  the  im- 
mediate neighborhood  and  on  both  sides  of  the  apparent  value. 
That  value  of  the  function  which  is  either  greater  or  less  than 
all  those  which  immediately jJrecede  or  follow  is  the  value  desired. 

PROBLEMS 

1.  A  line  10  in.  long  is  divided  into  two  segments  which 
are  taken  as  the  base  and  altitude  of  a  rectangle,  (a)  Express 
the  area  of  the  rectangle  as  a  function  of  one  of  the  segments, 
(b)  Plot  this  function,  (c)  Discuss  the  increase  and  decrease  of 
area  as  the  length  of  one  segment  changes  from  0  to  10  in. 
(d)  What  length  of  segment  gives  a  maximum  area  ?  (e)  What 
is  the  maximum  area  ?    (/)  Is  there  a  minimum  area  ? 

Suggestion.   Let  x  =  one  segment. 

10  —  a;  =  other  segment. 
X  (10  —  x)  =  area. 


FUNCTIONALITY  95 

2.  Express  the  sum  of  a  variable  number  and  its  reciprocal 
as  a  function  of  the  number.  Plot  the  function  and  investigate 
for  regional  changes.  What  is  the  minimum  value  of  the  sum 
of  a  number  and  its  reciprocal  ? 

3.  An  open-top  tank  with  a  square  base  is  to  be  built  to 
contain  32  cu.  ft.  What  should  be  the  dimensions  in  order  to 
require  the  smallest  amount  of  steel  plate  for  construction  ? 

Suggestion.   Let  a;  =  a  side  of  the  base. 

32 
Then  — -  =  depth  of  the  tank. 

x^  H =  surface  of  the  tank. 

X 

128 
Plot  the  function  x^  H and  determine  x  for  the  minimum  value. 

X. 

4.  Express  the  area  of  a  variable  rectangle  inscribed  in  a 
circle  whose  radius  is  4  in.,  as  a  function  of  the  base.  What 
are  the  dimensions  of  the  rectangle  of  greatest  possible  area  ? 

Suggestion.  Make  a  drawing  of  the  circle  and  rectangle  and 
note  how  the  area  changes  as  the  base  of  the  rectangle  increases 
from  0  to  8  in.  A  diagonal  of  the  rectangle  is  a  diameter  of  the 
circle.   Why  ? 

Let  X  =  base  of  the  rectangle. 

Then  V64  —  x^  =  altitude  of  the  rectangle. 

X  V64  —  x^  =  area  of  the  rectangle. 
Plot  this  function  and  determine  the  value  of  x  that  makes  it  a 
maximum. 

5.  Show  that  the  largest  rectangle  having  a  perimeter  of 
24  in.  is  a  square. 

6.  What  are  the  dimensions  of  the  greatest  rectangle  in- 
scribed in  a  right  triangle  whose  base  is  12  in.  and  altitude 
8  in.  ? 

7.  From  the  cube  of  a  variable  number  six  times  the  num- 
ber is  subtracted.  What  value  of  the  variable  would  make  this 
function  a  minimum  ?   Discuss  the  functional  variation  in  full 


96 


APPLIED  MATHEMATICS 


8.  From  a  variable  number  its  logarithm  is  subtracted. 
What  value  of  the  variable  number  would  make  this  difference 
a  minimum  ? 

9.  Two  towns  A  and  B  (Fig.  53) 
are  3  and  4  mi.  respectively  from 
the  shore  of  a  lake  CD.  If  CD  is 
a  straight  line  7  mi.  long,  where 
must  a  pumping  station  P  be  built 
to  supply  the  towns  with  water  with 
the  least  amount  of  pipe  ? 

10.  If  t  represents  the  number  of  tons  of  coal  used  by  a 
steamer  on  a  trip,  and  v  represents  the  speed  of  the  boat  per 
hour,  the  following  relation  holds :  ^  =  .3  +  .001  v^.  Other  ex- 
penses are  represented  by  one  ton  of  coal  per  hour.  What  speed 
would  make  the  cost  of  a  1000-mi.  trip  a  minimum  ? 

11.  The  cost  of  an  article  is  35  cents.  If  the  number  sold  at 
different  prices  is  given  by  the  following  table,  find  the  selling 
price  which  would  probably  give  the  greatest  profit. 


Selling  price  in  dollars 
Number  articles  sold  . 


.50 
3600 


.60 
3100 


.75 
2640 


.90 
2080 


1.00 
1300 


1.10 
700 


Suggestion.  First  from  the  given  table  plot  a  curve  to  show  the 
probable  number  sold  at  prices  from  50  cents  to  $1.10.  Then  on 
the  same  axes  with  different  vertical  units  plot  the  curve  to  show 
the  profits  at  the  various  prices.  J^rofit  =  (selling  price  —  cost)  x 
number  sold. 

To  determine  the  turning  point  of  the  second  curve  somewhat 
closely  it  will  be  necessary  to  locate  intermediate  points ;  e.g.  for 
the  selling  price  at  80  cents  and  85  cents.  The  number  probably 
sold  at  these  prices  may  be  found  from  the  first  curve. 

12.  Devise  other  problems  in  maxima  and  minima  and 
solve  them. 


CHAPTER  IX 

EXERCISES  FOR  ALGEBRAIC  SOLUTION  IN  PLANE  GEOMETRY 

67.  During  the  year  given  to  plane  geometry  these  exercises 
not  only  serve  as  a  review  of  algebra,  but  they  should  also 
develop  in  the  pupils  an  ability  to  attack  successfully  many 
geometrical  problems  from  the  algebraic  side.  The  figures  for 
the  first  exercises  should  be  carefully  drawn  with  ruler,  com- 
passes, and  protractor,  and  the  drawing  should  check  the 
algebraic  work.  Later  the  figures  may  be  sketched.  The  num- 
bers and  letters  should  be  j)ut  on  the  given  and  required  parts 
in  the  drawing,  and  the  equations  set  up  from  the  figures. 
Represent  lines,  angles,  and  areas  by  a  single  small  letter. 
Check  all  results. 

Complementary  and  Supplementary  Angles 

1.  Find  two  complementary  angles  whose  difference  is  (a) 
20°;  (b)  52°;  (c)  5°  8'  10";  ((T)  x". 

2.  x/2  and  x/S  (x  +  40  and  x  —  30)  are  complementary 
angles.    Find  x  and  the  angles. 

3.  Find  the  angle  that  is  the  complement  of  (a)  8  times 
itself ;  (b)  7  times  itself ;  (c)  3  times  itself ;  (d)  n  times  itself. 

4.  How  iiftiny  degrees  are  there  in  the  complementary  angles 
which  are  in  the  ratio  (a)  1  :  2  ?  (J)  4  :  5  ?  (c)  3.5  :  6.5?  {d)m  :  n? 

5.  Find  the  value  of  two  supplementary  angles  if  one  is  9 
(15)  times  as  large  as  the  other. 

6.  How  many  degrees  are  there  in  an  angle  that  is  the  sup- 
plement of  (a)  4  times  itself  ?  (b)  7  times  itself  ?  (c)  ^  of  itself  ? 
(d)  n  times  itself  ? 

97 


98  APPLIED  MATHEMATICS 

7.  Of  two  supplementary  adjacent  angles,  one  lacks  7°  of 
being  10  times  as  large  as  the  other.  How  many  degrees  in 
each  ? 

8.  If  10°  (7°)  be  added  to  one  of  two  supplementary  angles 
and  20°  (8°)  to  the  other,  the  resulting  angles  will  be  in  the 
ratio  2  :  5  (3  :  4).    Find  the  angles. 

9.  If  6°  (5°)  be  taken  from  one  of  two  supplementary  angles 
and  added  to  the  other,  the  ratio  of  the  two  angles  thus  found 
is  2  :  7  (13  :  5).    What  are  the  angles  ? 

10.  To  one  of  two  supplementary  angles  add  11°  (9°)  and 
from  the  other  subtract  16°  (5°).  The  two  angles  thus  obtained 
will  be  to  each  other  as  3  :  4  (5  :  12).    Find  the  angles. 

11.  How  many  degrees  are  there  in  an  angle  whose  supple- 
ment is  (a)  5  times  its  complement  ?  (b)  |  of  its  complement  ? 
(c)  n  times  its  complement  ? 

12.  Find  the  angle  whose  supplement  and  complement  added 
together  make  112°  (208°). 

13.  If  3  (8)  times  the  complement  of  an  angle  be  taken  from 
its  supplement,  the  remainder  is  10°  (76°).    Find  the  angle. 

14.  If  3  times  an  angle  added  to  5  times  its  supplement 
equals  20  times  its  complement  (supplement),  what  is  the 
angle  ? 

15.  The  angles  formed  by  one  line  meeting  another  are  in 
the  ratio  7  :  11  (3  :  8).    How  many  degrees  in  each  ? 

16.  Construct  a  graph  to  show  the  complement  of  any  angle. 
(Take  a  large  square  each  way  equal  10°.  Locate  a  few  points  : 
X  =  10,  ?/  =  80 ;  X  =  40,  ?/  =  50 ;  x  =  90,  y  =  0 ;  and  draw  a 
straight  line  through  them.)    What  is  the  equation  of  this  line  ? 

17.  On  the  same  sheet  of  squared  paper  construct  a  graph  to 
show  the  supplement  of  any  angle.  What  is  the  equation  of 
the  straight  line  ? 

18.  On  the  same  sheet  of  squared  paper  as  in  the  last  two 
problems  draw  a  straight  line  from  (x  =  0,  y  =  0)  to  (x  =  80°, 


EXERCISES  FOR  ALGEBRAIC  SOLUTION  99 

y  =  160°).  Read  off  a  few  pairs  of  angles  given  by  points  on 
this  line.  What  is  the  equation  of  this  line  ?  On  this  line 
mark  the  points  that  answer  the  question,  If  one  of  two  comple- 
mentary (supplementary)  angles  is  twice  the  other,  how  many 
degrees  in  each  ? 

19.  Find  two  complementary  angles  such  that  the  sum  of 
twice  one  and  3  times  the  other  is  210°.    Solve  graphically. 

20.  Two  complementary  angles  are  in  the  ratio  2  :  3  (7  :  8). 
Find  the  number  of  degrees  in  each.    Solve  graphically. 

21.  Three  angles  make  up  all  the  angular  magnitude  about  a 
point.  The  difference  of  the  first  and  second  is  10°  (20°),  and 
of  the  second  and  third  is  100°  (2°).  How  many  degrees  in  each 
angle  ? 

22.  The  sum  of  four  angles  about  a  point  is  360°.  The 
second  is  3  times  the  first,  the  third  is  10°  greater  than  the  sum 
of  the  first  and  second,  and  the  fourth  is  twice  the  first.  Find 
the  angles. 

23.  Of  the  angles  formed  by  two  intersecting  lines,  one  is 
5  (3^)  times  another.    What  are  the  angles  ? 

Parallel  Lines 

24.  Two  parallels  are  cut  by  a  transversal  making  one  ex- 
terior angle  3  (5f )  times  the  other  exterior  angle  on  the  same 
side  of  the  transversal.    Find  all  the  angles. 

25.  If  two  parallels  are  cut  by  a  transversal  making  two  ad- 
jacent angles  differ  by  20°  (36°  20'),  find  all  the  angles. 

26.  If  a  transversal  of  two  parallels  makes  the  sum  of  5  (4) 
times  one  interior  angle  and  2  (3)  times  the  other  interior 
angle  on  the  same  side  of  the  transversal  equal  to  420°  (625°), 
find  all  the  angles. 

27.  The  sum  of  one  pair  of  alternate-interior  angles  formed 
by  a  transversal  of  two  parallels  is  8  (6J)  times  the  sum  of  the 
other  pair.    Find  all  the  angles. 


100  APPLIED  MATHEMATICS 


Triangles 


28.  Of  the  angles  of  a  triangle  the  second  is  twice  the  first, 
and  the  third  is  3  times  the  second.  How  many  degrees  in 
each  angle  ? 

29.  Find  the  angles  of  a  triangle  ABC,  given : 

(a)  A  3  times  and  B  4  times  as  large  as  C. 

(b)  A  3  times  as  large  as  C  and  B  ^  oi  C. 

(c)  AW  and  B  25°  smaller  than  C. 

(d)  A:B:  C  =  2  :  3  :  4(3  :  5  :  10). 

30.  In  a  triangle  ABC  angle  ^  is  6  times  angle  B,  and  angle 
C  is  ^  of  angle  A.    Find  the  three  angles. 

31.  Find  the  angles  of  the  triangle  ABC  when  A  is  43°  more 
than  f  of  B,  and  B  is  18°  less  than  4  times  C. 

32.  The  sum  of  the  first  and  second  angles  of  a  triangle  is 
twice  the  third  angle,  and  the  third  angle  added  to  3  times  the 
second  equals  140°  less  the  third  angle.   Find  the  three  angles. 

33.  In  a  triangle  the  sum  of  twice  the  first  angle,  3  times  the 
second,  and  the  third  is  320° (400°);  and  the  sum  of  the  first, 
twice  the  second,  and  3  times  the  third  is  440°  (310°).  Find  the 
angles. 

34.  In  a  triangle  ABC,  A  lacks  106°  of  being  equal  to  the 
sum  of  B  and  C,  and  C  lacks  10°  of  being  equal  to  the  sum  of 
A  and  B.    Find  the  angles. 

35.  The  vertical  angle  of  an  isosceles  triangle  is  68°.  Find 
the  base  angles. 

36.  One  base  angle  of  an  isosceles  triangle  is  25°  (47°).  Find 
the  vertical  angle. 

37.  Find  the  angles  of  an  isosceles  triangle  if  a  base  angle  is 
4(5)  times  the  vertical  angle. 

38.  In  an  isosceles  triangle  the  vertical  angle  is  36°  (75°) 
larger  than  a  base  angle.    Find  the  angles. 

39.  In  an  isosceles  triangle  5  times  a  base  angle  added  to  3 
times  the  vertical  angle  equals  490°  (530°).    Fiiid  the  angles. 


EXERCISES  FOR  ALGEBRAIC  SOLUTION         101 

40.  Find  the  angles  of  an  isosceles  triangle  in  which  the  ex- 
terior angle  at  the  base  is  95°  (140°). 

41.  TRe  angle  at  the  vertex  of  an  isosceles  triangle  is  ^  (^)  of 
the  exterior  angle  at  the  vertex.  Find  the  angles  of  the  triangle. 

42.  A  base  angle  of  an  isosceles  triangle  is  12  (n)  times  the 
vertical  angle.    Find  the  angles  of  the  triangle. 

'  43.  What  are  the  angles  of  an  isosceles  triangle  in  which  the 
vertical  angle  is  12°  more  than  ^(|)  of  the  sum  of  the  base 
angles  ? 

44.  Construct  a  graph  to  show  the  change  in  the  vertical 
angle  y  of  an  isosceles  triangle  as  a  base  angle  x  increases  from 
0°  to  90°. 

45.  The  vertical  angle  of  an  isosceles  triangle  lacks  8°  (20°) 
of  being  ^^  (.9)  of  a  right  angle.    Find  all  the  angles. 

46.  The  acute  angles  of  a  right  triangle  are  x  and  2x(3y 
and  5  y).   Find  them. 

47.  The  difference  of  the  acute  angles  of  a  right  triangle  is 
18°  (37°).    Find  them. 

48.  If  the  acute  angles  of  a  right  triangle  are  in  the  ratio 
(a)  2  :  3,  (6)  7  :  8,  (c)  m  :  n,  find  the  angles. 

49.  In  a  right  triangle  the  sum  of  twice  one  acute  angle  and 
3  times  the  other  is  211°  (192°).    Find  the  angles. 

Polygons 

50.  How  many  sides  has  a  polygon  the  sum  of  whose  inte- 
rior angles  is  720°  (2340°)? 

51.  An  interior  angle  of  a  regular  polygon  is  165°  (160°). 
How  many  sides  has  the  polygon  ? 

52.  How  many  sides  has  a  polygon  the  sum  of  whose  inte- 
rior angles  equals  2  (12)  times  the  sum  of  the  exterior  angles  ? 

53.  How  many  sides  has  a  polygon  the  sum  of  whose  interior 
angles  exceeds  the  sum  of  the  exterior  angles  by  1080°  (2700°)  ? 


102  APPLIED  MATHEMATICS 

54.  Construct  a  graph  to  show  the  sum  of  the  angles  of  a 
polygon  as  the  number  of  sides  increases  from  3  to  12. 

55.  Construct  a  graph  to  show  the  number  of  degrees  in 
each  angle  of  a  regular  polygon  of  n  sides  for  values  of  n  from 
3  to  36. 

56.  If  the  number  of  sides  of  a  regular  polygon  be  increased 
by  2(3),  each  of  its  interior  angles  is  increased  by  15°  (10°). 
How  many  sides  has  the  polygon  ? 

57.  By  how  many  must  the  number  of  sides  of  a  regular 
polygon  of  12(15)  sides  be  increased  in  order  that  each  inte- 
rior angle  may  be  increased  18°  (6°)? 

58.  By  how  many  must  the  number  of  sides  of  a  regular 
polygon  of  8(20)  sides  be  increased  if  each  exterior  angle  is 
diminished  5°  (6°)  ? 

59.  Construct  a  curve  to  show  the  number  of  degrees  in  an 
exterior  angle  of  a  regular  polygon  as  the  number  of  sides 
increases  from  3  to  18. 

60.  The  perimeter  of  a  triangle  is  176(50.4)  ft.  in  length 
and  the  sides  are  as  1 :  3  :  4(2  :  5  :  7).    Find  the  sides. 

61.  The  perimeter  of  a  triangle  bears  to  one  side  the  ratio 
3  : 1  (15  :  4)  and  to  another  side  the  ratio  4  : 1  (5  :  2).  What 
part  of  the  perimeter  is  the  third  side  ? 

62.  The  sum  of  the  three  sides,  a,  b,  and  c,  of  a  triangle  is 
35  ft. ;  twice  a  is  5  ft.  less  than  the  sum  of  b  and  c,  and  twice 
c  is  4  ft.  more  than  the  sum  of  a  and  b.    Find  each  side. 

63.  If  the  perimeter  and  base  of  an  isosceles  triangle  are  in 
the  ratio  4  : 1  (5  :  2),  what  part  of  the  perimeter  is  one  of  the 
equal  sides  ? 

64.  Find  the  perimeter  of  an  isosceles  triangle  if  it  is  4  (8^) 
times  the  base,  and  one  of  the  equal  sides  is  4(55)  ft.  longer 
than  the  base. 

65.  In  an  isosceles  right  triangle  the  perpendicular  from 
the  vertex  to  the  hypotenuse  is  12  (30)  cm.  long.  How  long  is 
the  hypotenuse  ? 


EXERCISES  FOR  ALGEBRAIC  SOLUTION         103 

66.  If  the  hypotenuse  of  an  isosceles  right  triangle  is  26  (8)  in. 
long,  what  is  the  length  of  the  perpendicular  from  the  vertex  to 
the  hypotenuse  ? 

Parallelograms 

67.  One  angle  of  a  parallelogram  is  4(9)  times  its  consecu- 
tive angle.   Find  all  the  angles. 

68.  An  angle  of  a  parallelogram  is  3(2§)  times  one  of  the 
other  angles.    Find  all  the  angles. 

69.  Find  the  angles  of  a  parallelogram  if  the  difference  of 
two  consecutive  angles  is  20°  (90°). 

70.  If  two  consecutive  angles  of  a  parallelogram  are  in  the 
ratio  17  : 1  (4  :  5),  how  many  degrees  in  each  angle  ? 

71.  How  many  degrees  in  each  angle  of  a  parallelogram  when 
an  angle  exceeds  ^  (^)  of  its  consecutive  angle  by  30°  (56°)  ? 

72.  The  number  of  degrees  in  one  angle  of  a  parallelogram 
equals  ^  of  the  square  of  the  number  of  degrees  in  the  con- 
secutive angle.    Find  all  the  angles. 

73.  Prove  algebraically  that  if  two  angles  x  and  y  of  a  quad- 
rilateral are  supplementary,  the  other  two  angles  a  and  h  are 
also  supplementary. 

74.  Find  the  sides  of  a  parallelogram  if  one  side  is  ^  (|)  of 
another  side  and  the  perimeter  is  28  (84)  cm. 

75.  One  side  of  a  parallelogram  is  4  (5)  in.  longer  than  an- 
other side  and  the  perimeter  is  36  (58)  in.    Find  the  sides. 

76.  The  sum  of  two  adjacent  sides  of  a  rhomboid  is  |  (f )  of 
the  difference  of  those  sides.  Find  the  sides  if  the  perimeter 
is  18.3(82)  cm. 

77.  One  angle  of  a  rhombus  is  60°.  If  5  (2)  times  the  perim- 
eter exceeds  the  square  of  the  shorter  diagonal  by  19(13|), 
find  a  side  of  the  rhombus. 

78.  In  a  rhomboid  two  of  whose  sides  are  a  and  i,  3  times 
a  exceeds  twice  i  by  11,  and  the  sum  of  twice  a  and  5  times  h 
is  20.    Find  the  perimeter. 


104  APPLIED  MATHEMATICS 

79.  In  one  of  the  triangles  formed  by  the  diagonals  of  a 
rhombus  and  one  of  the  sides  of  the  rhombus  the  two  smaller 
angles  are  in  the  ratio  2  :  3(1 :  3).  Find  all  the  angles  of  the 
rhombus. 

80.  The  perimeter  of  a  parallelogram  is  16(9.6),  and  the 
square  of  one  side  added  to  4  (2)  times  an  adjacent  side  equals 
37  (8.6).    Find  the  sides  of  the  parallelogram. 

81.  In  a  rhombus  one  of  whose  angles  is  60°  the  shorter 
diagonal  is  10  in.  (5  ft.  6  in.).    Find  the  perimeter. 

82.  Two  sides  of  a  rectangle  are  x  and  x^i^x  and  7  a;)  and 
the  perimeter  is  60(40).    Find  the  sides. 

Circles 

83.  The  circumference  of  a  circle  is  divided  into  three  parts. 
Find  the  number  of  degrees  in  each  part  if  the  second  contains 
3(6)  times  as  many  as  the  first  part,  and  the  third  part  con- 
tains 5  (7)  times  as  many  as  the  first  part. 

84.  In  a  circle  a  diameter  and  a  chord  are  drawn.  The 
diameter  is  4  (5)  in.  longer  than  the  chord  and  the  diameter 
and  chord  together  are  18(20)  in.  long.    How  long  is  each  ? 

85.  There  are  100°  (a;°)  in  one  of  the  arcs  subtended  by  a 
chord.    How  many  degrees  are  there  in  the  other  arc  ? 

86.  In  one  of  the  arcs  subtended  by  a  chord  there  are 
50°  (120°)  more  than  in  the  other  arc.  How  many  degrees 
in  each  arc  ? 

87.  Find  the  side  of  a  square  inscribed  in  a  circle  whose 
radius  is  30  (42.5)  mm. 

88.  A  triangle  whose  perimeter  is  36  (72)  mm.  is  inscribed 
in  a  circle.  The  first  side  is  ^  of  the  second  and  f  of  the  third. 
Find  the  three  sides. 

89.  In  a  circle  of  radius  8  (12)  in.  a  chord  is  drawn  equal 
in  length  to  the  radius.   How  far  is  it  from  the  center? 


EXERCISES  FOR  ALGEBRAIC  SOLUTION  105 

90.  A  circle  containing  280(308)  sq.  ft.  is  divided  into  three 
parts  by  radii.  The  third  part  equals  twice  the  second,  and 
the  second  part  is  20  sq.  ft.  larger  than  the  first.  Find  the 
ai'ea  of  each  part. 

91.  A  line  1  (3.6)  ft.  long  intersects  a  circumference  in  two 
points.  If  the  part  inside  the  circumference  is  twice  the  length 
of  the  part  outside,  how  long  is  the  part  which  forms  the 
chord  ? 

92.  A  number  of  coins  are  placed  in  a  row  touching  one 
another,  and  the  length  of  the  row  is  measured.    3  quarters, 

2  nickels,  and  5  dimes  measure  204  mm. ;  1  quarter,  3  nickels, 
and  2  dimes  measure  123  mm. ;  and  1  quarter,  1  nickel,  and 
1  dime  measure  63  mm.  Find  the  diameter  of  each  coin.  Check. 

93.  A  boy  has  20  copper  disks  ;  part  of  them  are  20  mm.  in 
diameter  and  the  rest  are  30  mm.  The  sum  of  their  diameters 
is  520  mm.    How  many  of  each  kind  has  he  ? 

94.  Two  diameters  are  drawn  in  a  circle,  making  at  the 
center  one  of  the  supplementary  adjacent  angles  3  times  the 
other.    How  many  degrees  in  each  angle  ? 

95.  A  chord  6  (4)  in.  long  is  4  (6)  in.  from  the  center  of  a 
circle.    Find  the  radius  of  the  circle. 

96.  A  chord  16  (4)  in.  long  is  at  a  distance  of  6  (8)  in.  from 
the  center  of  a  circle.    What  is  the  length  of  a  chord  which  is 

3  (1)  in.  from  the  center  ? 

97.  A  chord  8(12)  in.  long  bisects  at  right  angles  a  radius. 
How  long  is  the  radius  ? 

98.  The  radius  of  a  circle  is  5  (3)  in.  How  far  from  the 
center  is  a  chord  8(4)  in.  long  ? 

99.  The  radius  of  a  circle  is  r.  What  is  the  length  of  a 
chord  whose  distance  from  the  center  is  \  (J)  r  ? 

100.  Find  the  length  of  the  longest  and  shortest  chords  that 
can  be  drawn  through  a  point  9  (6)  in.  from  the  center  of  a 
circle  whose  radius  is  15  (8)  in. 


106  APPLIED  MATHEMATICS 

101.  The  sum  of  the  longest  and  the  shortest  chords  through 
a  point  3  (8)  in.  from  the  cente'r  of  a  circle  is  18  (64)  in.  Find 
the  radius  and  the  two  chords. 

102.  Construct  a  curve  to  show  the  length  of  a  chord  in  a 
circle  of  radius  8  in.  as  the  distance  of  the  chord  from  the 
center  increases  from  0  to  8  in. 

103.  A  circle  is  circumscribed  about  a  right  triangle  whose 
legs  are  6  and  8  (5  and  12)  in.    Find  the  radius  of  the  circle. 

104.  The  legs  of  a  right  triangle  inscribed  in  a  circle  are 
6  a;  and  12-a;  {x  and  3  a;)  and  the  radius  of  the  circle  is  13(5)  in. 
Find  the  sides  of  the  triangle. 

105.  From  the  point  of  tangency  P,  a  distance  PA  equal  to 
twice  the  radius  is  measured  oif  on  the  tangent.  If  the  distance 
from  A  to  the  center  of  the  circle  is  10(6)  in.,  find  the  radius. 

106.  In  a  circle  of  radius  8  (5)  in.  two  parallel  chords  lie  on 
opposite  sides  of  the  center.  One  is  twice  as  far  from  the  center 
as  the  other.  If  the  sum  of  the  squares  of  the  half  chords  is 
123(10)  in.,  find  the  distance  each  chord  is  from  the  center. 

107.  The  perimeter  of  an  -.inscribed  isosceles  trapezoid  is 
38(88)  in.  One  of  the  parallel  sides  is  |(.7)  of  the  other  and 
one  of  the  nonparallel  sides  is  9^  (30)  in.  shorter  than  the 
longest  side  of  the  trapezoid.    Find  each  side. 

108.  Two  circles  touch  each  other  and  their  centers  are 
8  (a)  in.  apart.  The  diameter  of  one  is  10  (d)  in.  What  is  the 
diameter  of  the  other  ? 

109.  Two  circles  are  tangent  externally.  The  difference  of 
their  radii  is  8  (a)  in.  and  the  distance  between  their  centers  is 
12  (h)  in.    Find  the  radii. 

110.  The  distance  between  the  centers  of  two  circles  is 
18(a)  in.,  which  is  one  half  the  sum  of  their  radii.  Find  the 
radii. 

111.  One  angle  of  an  inscribed  triangle  is  35° (50°)  and  one 
of  its  sides  subtends  an  are  of  113°  (150°).  Find  the  other 
angles  of  the  triangle. 


EXERCISES  FOR  ALGEBRAIC  SOLUTION  107 

112.  The  circumference  of  a  circle  is  divided  into  three  arcs 
in  the  ratio  1 :  2  :  3(2  :  3  :  5).  Find  the  angles  of  the  triangle 
formed  by  the  chords  of  the  arc. 

113.  A  triangle  is  inscribed  in  a  circle.  The  sum  of  the  first 
and  third  angles  is  twice  the  second  angle,  and  the  difference 
of  the  first  and  second  is  20°.  How  many  degrees  in  each  of 
the  three  arcs  ? 

114.  Construct  a  graph  to  show  the  change  in  an  inscribed 
angle  y,  as  the  arc  intercepted  by  its  sides  increases  from 
0  to  180°. 

115.  An  isosceles  triangle  is  inscribed  in  a  circle.  The  number 
of  degrees  in  the  arc  upon  which  the  vertical  angle  stands  is 
8(3^)  times  the  number  of  degrees  in  a  base  angle  of  the 
triangle.    Find  the  angles  of  the  triangle. 

116.  Consecutive  sides  of  an  inscribed  quadrilateral  subtend 
arcs  of  82°,  99°,  67°,  and  x°  respectively.  Find  each  angle  of 
the  quadrilateral;  also  each  of  the  eight  angles  formed  by  a 
side  and  a  diagonal. 

117.  How  many  degrees  in  each  angle  of  a  quadrilateral 
inscribed  in  a  circle,  if  the  sides  subtend  arcs  which  are  in 
the  ratio  1:2:  3:4(2:3:5:  6)? 

118.  A  right  triangle  is  inscribed  in  a  circle.  If  one  acute 
angle  of  the  triangle  is  §  (|)  of  the  other,  how  many  degrees  in 
each  of  the  three  arcs  ? 

119.  ABCD  is  an  inscribed  trapezoid.  If  the  angle  A  is  twice 
angle  C,  find  each  angle. 

120.  Two  chords  AB  and  CD  intersect  within  a  circle  at  P. 
The  angle  APC  is  50°,  arc  DB  is  40°,  and  arc  AD  is  160°. 
Find  the  other  arcs  and  angles. 

121.  Two  chords  AB  and  CD  intersect  within  a  circle  at  P. 
Arc  BD  is  twice  arc  AC,  and  arc  CB  is  twice  arc  DA.  Angle 
DP  A  is  twice  angle  APC.    Find  the  arcs  and  angles. 

122.  The  angle  y  is  formed  by  two  chords  AB  and  CD  inter- 
secting in  a  circle,  and  the  two  intercepted  arcs  A  C  and  DB  ai'e 


108  APPLIED  MATHEMATICS 

90°  and  x°  respectively.  What  is  the  equation  connecting  y 
and  X  ?  Construct  a  graph  to  show  the  change  in  y  as  a; 
increases  from  0  to  90°. 

123.  From  a  point  without  a  circle  two  secants  are  drawn, 
making  one  of  the  intercepted  arcs  3(5)  times  the  other.  If 
the  sum  of  the  other  two  arcs  is  200°  (300°),  what  is  the  angle 
formed  by  the  secants  ? 

124.  The  angle  y  is  formed  by  two  secants  intersecting  with- 
out a  circle.  The  intercepted  arcs  are  90°  and  a;°(a;<90). 
What  is  the  equation  connecting  y  and  x  ?  Construct  a  graph 
to  show  the  change  in  ?/  as  cc  increases  from  0  to  90°. 

125.  Two  tangents  drawn  from  an  exterior  point  to  a  circle 
make  an  angle  of  60°  (80°).  Find  the  two  arcs.  Join  the  points 
of  tangency  and  find  the  other  two  angles  in  the  triangle  thus 
formed. 

126.  Through  the  ends  of  an  arc  of  45°  (100°)  tangents  to 
the  circle  are  drawn.  Find  the  angle  formed  by  the  tangents. 
Find  the  other  two  angles  in  the  triangle  formed  by  joining 
the  points  of  tangency. 

127.  Find  the  angle  formed  by  two  tangents  to  a  circle  drawn 
from  a  point  at  a  distance  from  the  center  of  the  circle  equal 
to  the  diameter. 

128.  From  P,  a  point  without  a  circle,  two  tangents  PA  and 
PB,  and  a  secant  PC  are  drawn.  The  arc  vl^  equals  160° (100°). 
If  the  difference  of  the  angles  BPC  and  CPA  is  10° (25°),  find 
the  angles. 

129.  From  a  point  without  a  circle  of  radius  4  (8)  in.  a 
secant  through  the  center  and  a  tangent  are  drawn.  If  the 
angle  formed  by  the  secant  and  tangent  is  30°  (60°),  find 
the  distance  from  the  point  to  the  center  of  the  circle,  and 
the  length  of  the  tangent. 

130.  In  an  equilateral  triangle  whose  sides  are  40  (60)  mm. 
a  circle  is  inscribed.  Find  the  radius  of  the  circle.  Find  the 
radius  of  the  circijmscribed- circle. 


EXERCISES  FOR  ALGEBRAIC  SOLUTION  109 

Ratio 

131.  Express  the  ratio  of  the  following  pairs  of  numbers  in 
the  simplest  form : 

(a)  168  and  252.  (A)  148 x^  and  185a;*. 

(b)  387  and  602.  (i)  x"  +  5x  +  6  smd  x  +  S. 
(  c  )  I  and  |.  0" )  a;2  +  2  cc  —  15  and  x  +  5. 

(d)  6^and30|.  (k)  x^ +  3x  +  2siudx^+ 4.X  +  S. 

(e)  Handf.  (0  x^  +  6x  +  5iindx''-\-Sx  +  W. 
(/)  .125  and  3.75.  ^^^^  -f|  ^^^^  f+6^  +  8  . 
(g)  Qa^x  and  30a*a;.  ^    ^  x  +  3          x"  +  1  x  +  12 

132.  Squares  are  constructed  on  the  lines  <i  and  h.  Find  the 
ratio  of  the  areas  : 

(a^  a  =  5  in.,  b  =  10  in.  (c)  a  =  4  cm.,  b  =  12  cm. 

(b)  a  =  3^  in.,  b  =  7  in.  (d)  a  =  14  mm.,  b  =  35  cm. 

133.  On  a  sheet  of  squared  paper  let  the  bottom  line  be  the 
aj-axis  and  the  left  border  line  be  the  ?/-axis,  and  the  side  of  a 
square  each  way  =  1.  Draw  a  straight  line  through  the  points 
(0,  0)  and  (8,  16).  Make  a  table  of  corresponding  values  of  x 
and  y.  What  is  the  ratio  of  y  to  a;  ?  What  is  the  equation  of 
the  line  ? 

134.  The  width  ?/  of  a  field  is  to  be  made  f  of  the  length  x. 
What  is  the  equation  connecting  y  and  x  ?  Construct  a  graph 
to  show  the  width  of  the  field  for  a  length  from  10  to  100  rd. 

135.  If  the  ratio  of  ?/  to  a?  is  2  :  3,  construct  a  graph  to  show 
the  relation.    What  is  the  equation  of  the  straight  line  ? 

136.  If  14  a;  —  9  2/  =  2  a-  —  ?/,  find  the  ratio  oix:y.  Construct 
the  graph. 

137.  What  is  the  ratio  ofa;:?/,  if7a;  —  6?/  =  3a;  +  4?/? 

138.  If  a; :  2/  =  4  :  5,  find  the  value  of  the  ratio  2x  +  y:l  x  —  y. 
Construct  the  graph. 

139.  Find  the  value  of  the  ratio  3  x"^  -\- 2  y"^ :  xy  +  y^,  \ix:y 
=  1:2. 


110  APPLIED  MATHEMATICS 

Proportion 

140.  Test  the  correctness  of  the  following  proportions ; 

^^  225      45'  ^^^         a^-b''        ~  a-b 

_87_^111  a;^+7g;  +  10_a;  +  2 

^^^  203  "259"  ^''^       {x  +  by      ~x  +  ^ 

141.  Find  x  in  the  following  proportions  : 
,  ,  18      32  a2  _  j2      ^  _  ^ 

28      35  .    .        a;  1 


42 

,  ,,  1.25   120 

90' 

^^>  .26  -  24  • 

96 
45' 

,  ,  a^  +  2ab  +  b'' 

111 

259 

a;2+7a;  +  10 
^J)      (x  +  ^y 

w-=?^-        (O 


X        18  ^    ^  a^'-b^      a-b 

^  _  16  g''  +  10  g  +  25  _  x 

^'^4.8~.24'  ^-^^  X  ~9' 

142.  What  number  can  be  added  to  7,  12,  1,  and  3  (5,  19, 
16,  and  52)  so  that  the  resulting  numbers  will  form  a  pro- 
portion ? 

,  143.  Find  the  nimibers  proportional  to  1,  2,  3,  4  (2,  5,  1,  3) 
that  may  be  added  regularly  to  5,  10,  15,  40  (11,  20,  8,  14)  so 
as  to  form  a  proportion. 

144.  The  line  joining  the  mid-points  of  the  nonparallel  sides 
of  a  trapezoid  is  20  (42)  in.  long.  Find  the  bases  if  one  is 
§  (.4)  of  the  other. 

145.  In  a  triangle  ABC  the  line  PQ  parallel  to  BC  divides 
the  side  A C  in  the  ratio  3  :  4  (5  :  9).  If  ^5  =  20  (9.8)  in.,  find 
the  two  segments  of  AB. 

146.  The  sum  of  the  two  sides  of  a  triangle  is  45  (63)  in.  A 
line  parallel  to  the  third  side  cuts  off  from  the  vertex  segments 
10  and  8  (4  and  20)  in.  long.    Find  the  two  sides. 

147.  A  line  100  (6)  ft.  long  is  divided  into  parts  in  the  ratio 
1 :  2  :  3  :  4  (2  :  3  :  7).    Find  each  part. 


EXERCISES  FOR  ALGEBRAIC  SOLUTION  111 

148.  Three  lines  are  in  the  ratio  2  :  3  :  4  (2  : 1 :  6)  and  their 
fourth  proportional  is  30  (24).    Find  the  length  of  each  line. 

149.  The  sum  of  two  sides  of  a  triangle  is  20  (5)  in.  The 
third  side,  18  (4^)  in.  long,  is  a  third  proportional  to  the  other 
two  sides.    Find  them. 

150.  One  side  of  a  triangle  is  2  in.  longer  than  the  first  side, 
and  the  third  side  is  5  in.  longer  than  the  first.  If  one  side  is 
a  mean  proportional  between  the  other  two,  find  the  three  sides. 

151.  The  three  sides  of  a  triangle  are  x,  y,  and  3.  The  cor- 
responding sides  of  a  similar  triangle  are  10,  20,  and  15.  Find 
X  and  y. 

152.  The  sum  of  the  three  sides  of  a  triangle,  x,  y,  and  z,  is 
15,  and  the  corresponding  sides  of  a  similar  triangle  are  a;  -(-  3, 
y  +  7,  and  «  +  5.    Find  the  sides  of  each  triangle. 

153.  The  three  sides  of  a  triangle  are  3x,  6x,  and  8  a; 
(x,  X  -{-1,  X  -\-  2),  and  the  corresponding  sides  of  a  similar  tri- 
angle are  3a;^,  6x^,  and  Sx^  (x^,  x^  -{-  x,  and  x^  +  2x).  If  the 
sum  of  the  perimeters  of  the  two  triangles  is  102  (75),  find  the 
sides  of  each  triangle. 

154.  The  sides  of  a  triangle  are  5, 8, 12  (12, 16, 20)  in.  Find  the 
segments  of  each  side  made  by  the  bisector  of  the  opposite  angle. 

155.  The  sum  of  two  sides  of  a  triangle  is  24  in.,  and  the 
bisector  of  the  included  angle  divides  the  third  side  into  parts 
4  and  8  in.  long.    Find  the  three  sides. 

156.  In  a  triangle  ABC,  AB  =  12  and  BC  =  36.  From  a 
point  on  ^5  at  a  distance  x  from  A  a  line  y  is  drawn  to  ^C 
parallel  to  the  base.  Construct  a  graph  to  show  the  length  of 
y  2kS,  X  increases  from  0  to  12, 

Right  Triangles 

157.  The  hypotenuse  of  a  right  triangle  is  8  in.  and  one 
angle  is  30°.  Find  (a)  the  other  two  sides ;  (V)  the  perpendic- 
ular from  the  vertex  of  the  right  angle  to  the  hypotenuse ;  (c) 
the  segments  of  the  hypotenuse. 


112  APPLIED  MATHEMATICS 

158.  One  leg  of  a  right  triangle  is  2  (3)  ft.  longer  than  the 
other  and  the  hypotenuse  is  4  (7)  ft.  longer  than  the  shorter 
leg.    Find  the  three  sides. 

159.  The  legs  of  a  right  triangle  are  12  and  16  (5  and  12) 
ft.  Find  (a)  the  hypotenuse ;  {h)  the  perpendicular  from  the 
vertex  of  the  right  angle  to  the  hypotenuse ;  (c)  the  segments 
of  the  hypotenuse. 

160.  The  perpendicular  from  the  vertex  of  the  right  angle 
of  a  right  triangle  to  the  hypotenuse  is  12  (3)  in.  long  and  the 
hypotenuse  is  26  (6.25)  in.  long.    Find  the  other  two  sides. 

161.  If  the  legs  of  a  right  triangle  are  a  and  h,  find  the  per- 
pendicular from  the  vertex  of  the  right  angle  to  the  hypotenuse, 
and  the  segm'ents  of  the  hypotenuse. 

162.  One  side  of  a  right  triangle  is  4.  Construct  a  curve  to  show 
the  length  of  the  hypotenuse  as  the  other  side  increases  from  0 
to  16.  (Let  the  bottom  line  be  the  ic-axis,  the  left  border  line  be 
the  2^-axis,  and  the  side  of  a  large  square  each  way  =  1.  Take  the 
side  4  on  the  vertical  axis  and  locate  the  points  of  the  curve 
with  compasses.    Check  a  few  of  the  points  by  computation.) 

Chords,  Tangents,  Secants 

163.  The  segments  of  a  chord  made  by  another  chord  are 
7  and  9(15  and  13)  in.,  and  one  segment  of  the  latter  chord 
is  3  (10)  in.    What  is  the  other  segment  ? 

164.  Two  chords  intersect,  making  the  segments  of  one  chord 
2  and  12(4  and  8)  in.,  and  one  segment  of  the  other  chord 
2  (14)  in.  longer  than  the  other  segment.    Find  the  two  chords. 

165.  One  of  two  intersecting  chords  is  14  (17)  in.  long,  and 
the  product  of  the  segments  of  the  other  chord  is  45  (60).  Find 
the  segments  of  the  lirst  chord. 

166.  Two  secants  intersect  without  a  circle.  The  external 
segment  of  one  is  20  (2)  in.  and  the  internal  segment  is  5  (4)  in. 
If  the  external  segment  of  the  other  secant  is  10(3)  in.,  find 
the  length  of  the  internal  segment. 


EXERCISES  FOR  ALGEBRAIC  SOLUTION      •  113 

167.  From  a  point  without  a  circle  two  secants  are  drawn 
whose  external  segments  are  5  and  6  (6  and  8)  in.  The  internal 
segment  of  the  former  is  13  (16)  in.  What  is  the  internal  seg- 
ment of  the  latter?  What  is  the  length  of  the  tangent  from 
the  same  point  ? 

168.  Two  secants  from  a  point  without  a  circle  are  24  in. 
and  22  in.  long.  If  the  external  segment  of  the  lesser  is  5  in., 
what  is  the  external  segment  of  the  greater  ?  What  is  the 
length  of  the  tangent  from  the  same  point  ? 

169.  A  tangent  and  a  secant  are  drawn  to  a  circle  from  an 
external  point.  The  external  and  internal  segments  of  the 
secant  are  respectively  2  (3)  in.  and  1  (4)  in.  shorter  than  the 
tangent.    What  is  the  length  of  the  tangent  ? 

170.  From  a  point  on  the  tangent  of  a  circle  6(15)  in.  from 
the  point  of  tangency  a  secant  is  drawn  whose  internal  seg- 
ment is  2(3)  times  the  external  segment.  Find  the  length  of 
the  secant. 

171.  A  tangent  intersects  a  secant  which  is  drawn  through 
the  center  of  a  circle.  The  length  of  the  tangent  is  4(t)  in., 
and  the  length  of  the  external  segment  of  the  secant  is  2(s) 
in.    Find  the  radius  of  the  circle  and  the  secant. 

172.  In  a  circle  of  radius  17  in.  a  point  P  is  taken  on  the 
diameter  15  in.  from  the  center.  What  is  the  product  of  the 
segments  of  chords  through  P?  Denoting  the  segments  by  x 
and  y,  what  is  the  equation  that  connects  x  and  ij  ?  In  this 
equation  give  values  to  x  and  make  a  table  of  values  of  x  and  y. 
Construct  a  curve  to  show  the  change  of  y  as  a;  increases  from 
2  to  32  in. 

173.  From  a  point  on  the  circumference  of  a  circle  of  9  in. 
diameter  a  tangent  6  in.  long  is  drawn.  From  the  end  of  the 
tangent  secants  are  drawn.  If  y  is  the  external  and  x  the  in- 
ternal segment  of  the  secant,  what  is  the  equation  connecting 
X  and  y  ?  Construct  a  curve  to  show  the  length  of  y  as  a;  in- 
creases from  0  to  9  in.  and  then  decreases  to  0. 


114  .  APPLIED  MATHEMATICS 


Area  of  Polygons 


174.  The  base  of  a  triangle  is  5(3)  times  the  altitude  and 
the  area  is  90  (75)  sq.  in.    Find  the  base  and  altitude. 

175.  The  area  of  a  triangle  is  130(42)  sq.  in.  and  the  altitude 
is  7  in.  less  (5  in.  more)  than  the  base.    Find  these  dimensions. 

176.  The  sum  of  the  base  and  altitude  of  a  triangle  is 
12(23)  in.  and  the  area  is  16(45)  sq.  in.  Find  the  base  and 
altitude. 

177.  Find  the  area  of  a  right  triangle  whose  base  is  20(32) 
and  the  sum  of  whose  hypotenuse  and  other  side  is  40(50). 

178.  The  altitude  of  an  equilateral  triangle  is  12(h)  ft.  Find 
its  sides  and  area. 

179.  The  altitude  of  a  triangle  is  16  in.  less  than  the  base. 
If  the  altitude  is  increased  3  in.  and  the  base  12  in.,  the  area 
is  increased  52  sq.  in.    Find  the  base  and  altitude. 

180.  If  the  hypotenuse  of  a  right  triangle  is  1  (8)  in.  longer 
than  one  leg,  and  8  (9)  in.  longer  than  the  other  leg,  what  is 
the  area  of  the  triangle  ? 

181.  If  the  area  of  an  equilateral  triangle  is  16  V3  (60)  sq.  in., 
jRnd  the  altitude  and  a  side. 

182.  If  a  denotes  the  area,  s  a  side,  and  h  the  altitude  of  an 
equilateral  triangle,  express  each  in  terms  of  the  others. 

183.  If  a  rectangle  is  7(8)  ft.  longer  than  it  is  wide  and 
contains  170(209)  sq.  ft.,  find  its  dimensions. 

184.  The  perimeter  of  a  rectangle  is  72(132)  ft.  and  its 
length  is  2  (5)  times  its  width.    Find  its  area. 

185.  A  rectangle  whose  length  is  8(5)  ft.  greater  than  3(4) 
times  its  width  contains  115  (3750)  sq.  ft.    Find  its  dimensions. 

186.  The  area  of  a  rectangle  is  36  sq.  ft.  Construct  a  curve 
to  show  the  altitude  as  the  base  increases  from  1  to  36  ft. 

187.  The  side  of  one  square  is  3  (4)  times  as  long  as  that  of 
another  square,  and  its  area  is  72  (90)  sq.  yd.  greater  than  that 
of  the  second  square.    What  is  the  side  of  each  square  ? 


EXERCISES  FOR  ALGEBRAIC  SOLUTION         115 

188.  One  side  of  a  square  is  3  (6)  yd.  less  than  2  (3)  times 
the  side  of  a  second  square,  and  the  difference  in  area  of  the 
squares  is  45  (756)  sq.  yd.    Find  the  area  of  each  square. 

189.  One  side  of  a  rectangle  is  10  (6)  ft.  and  the  other  side 
is  2(1)  ft.  longer  than  the  side  of  a  given  square.  The  area 
of  the  rectangle  exceeds  that  of  the  square  by  80  (174)  sq.  ft. 
Find  the  side  and  area  of  the  square. 

190.  The  floor  of  a  rectangular  room  contains  180  (240)  sq.  ft., 
and  the  length  of  the  molding  around  the  room  is  56  (62)  ft. 
Find  the  length  and  width  of  the  room. 

191.  A  picture  including  the  frame  is  10(9)  in.  longer  than 
it  is  wide.  The  area  of  the  frame,  which  is  3  (6)  in.  wide,  is 
192  (480)  sq.  in.    What  are  the  dimensions  of  the  picture  ? 

192.  The  dimensions  of  a  picture  inside  the  frame  are  12  in. 
by  16  in.  (5  in.  by  12  in.).  What  is  the  width  of  the  frame  if 
its  area  is  288  (138)  sq.  in.  ? 

193.  Around  a  square  garden  a  path  2  ft.  wide  is  made.  If 
376  sq.  ft.  are  taken  for  the  path,  find  a  side  of  the  garden. 

194.  Around  a  garden  100  ft.  by  120  ft.  a  man  wishes  to 
make  a  path  which  shall  occupy  /^  (J)  of  the  area.  How  wide 
must  the  path  be  made  ? 

195.  A  rectangular  building  having  a  perimeter  of  140  ft. 
is  inclosed  by  a  fence  whose  distance  from  the  building  is  ^  the 
width  of  the  building.  If  the  area  between  the  fence  and  build- 
ing is  1800  sq.  ft.,  find  how  far  the  fence  is  from  the  building. 

196.  An  open-top  box  is  made  from  a  square  piece  of  tin  by 
cutting  out  a  5  (2)-in.  square  from  each  corner  and  turning  up 
the  sides.  How  large  is  the  original  square  if  the  box  contains 
180(242)  cu.  in.? 

197.  An  open-top  box  is  formed  by  cutting  out  a  l(3)-in. 
square  from  each  corner  of  a  rectangular  piece  of  tin  2  (3)  times 
as  long  as  it  is  wide,  and  turning  up  the  sides.  If  the  total 
surface  of  the  box  is  284  (936)  sq.  in.,  find  the  dimensions  of 
the  piece  of  tin. 


116  APPLIED  MATHEMATICS 

198.  It  is  desired  to  make  an  open-top  box  from  a  piece  of 
tin  30  (24)  (15)  in.  sq.,  by  cutting  out  equal  squares  from  each 
corner  and  turning  up  the  strips.  What  should  be  the  length 
of  a  side  of  the  squares  cut  out  to  give  a  box  of  the  greatest 
possible  volume  ? 

Suggestion.   11  x=  side  of  square  cut  out,  volume  of  the  box  = 

Make  a  table  of  values  of  y,  giving  x  the  values  1,  2,  3  •  •  • . 
Locate  the  points  and  draw  a  smooth  curve  through  them.  The 
turning  point  of  the  curve  will  show  the  value  of  x  for  the 
greatest  volume. 

199.  Erom  a  rectangular  piece  of  tin  12  in.  by  24  in.  (16  in. 
by  36  in.)  it  is  desired  to  make  an  open-top  box  of  the  largest 
possible  volume,  by  cutting  out  equal  squares  from  the  corners 
and  turning  up  the  strips.  What  should  be  the  length  of  a  side 
of  the  squares  ? 

200.  The  altitude  of  a  trapezoid  is  5  (14)  in.,  the  area  is 
10(455)  sq.  in.,  and  the  difference  of  the  bases  is  2  (11)  in. 
Find  the  bases. 

201.  The  area  of  a  trapezoid  is  90  (495)  sq.  ft.,  the  line  join- 
ing the  mid-points  of  the  nonparallel  sides  is  6  (45)  ft.,  and  the 
difference  of  the  bases  is  6  (12)  ft.    Find  the  bases  and  altitude. 

202.  In  a  trapezoid  h  and  b'  are  the  bases,  h  the  altitude,  and 
a  the  area.    Find  each  in  terms  of  the  other. 

203.  The  base  of  a  triangle  is  12  in.  and  the  altitude  increases 
from  0  to  20  in.  Construct  a  graph  to  show  the  increase  in 
area  of  the  triangle. 

204.  The  base  and  altitude  of  a  triangle  increase  uniformly, 
and  the  altitude  is  always  twice  the  base.  Construct  a  curve 
to  show  the  change  in  the  area  of  the  triangle  as  the  base 
increases  from  0  to  10  ft. 

205.  The  base  and  altitude  of  a  triangle  are  24  in.  and  9  in. 
respectively.  What  is  the  area  of  the  triangle  formed  by  a  line 
parallel  to  the  base  and  6  (8)  (x)  in.  from  the  vertex  ? 


EXERCISES  FOR  ALGEBRAIC  SOLUTION         117 

206.  Ill  a  triangle  whose  base  is  12  in.  and  altitude  is  16  in. 
a  line  is  drawn  parallel  to  the  base  and  at  a  distance  x  from  the 
vertex.  If  y  =  the  area  of  the  triangle  cut  olf  from  the  vertex, 
what  is  the  equation  connecting  x  and  i/  ?  Construct  a  curve 
to  show  the  area  of  the  triangle  cut  off'  as  x  increases  from 
0  to  16  in. 

207.  The '  altitude  of  a  triangle  is  2  (3)  times  its  base. 
Through  the  mid-point  of  the  altitude  a  line  is  drawn  parallel 
to  the  base.  If  the  area  of  the  triangle  cut  off  is  36(5)  sq.  in., 
find  the  base  and  altitude  of  the  given  triangle. 

208.  The  sum  of  the  areas  of  two  similar  triangles  is 
240(290)  sq.  in.,  and  the  sides  of  one  are  2(2^)  times  the  cor- 
responding sides  of  the  other.    Find  the  area  of  each  triangle. 

209.  The  difference  of  the  areas  of  two  squares  is  39(324) 
sq.  ft.,  and  a  side  of  one  is  3  (14)  ft.  longer  than  a  side  of  the 
other.    Find  a  side  of  each  square. 

210.  The  sum  of  the  areas  of  two  squares  is  13(221)  sq.  ft., 
and  a  side  of  one  square  is  1  (9)  ft.  shorter  than  a  side  of  the 
other.    Find  a  side  of  each  square. 

211.  A  side  of  one  square  is  5(2)  in.  longer  than  a  side  of 
another  square,  and  the  areas  of  the  squares  are  in  the  ratio 
4  : 1  (16  :  9).    What  is  a  side  of  each  square  ? 

212.  Construct  a  curve  to  show  the  area  of  a  square  as  its 
sides  increase  from  0  to  13  in. 

Circles  and   Inscribed  Polygons 

213.  Construct  a  curve  to  show  the  area  of  a  circle  as  its 
radius  increases  from  0  to  16  in.  (Locate  points  for  r  =  0,  2,  4, 
...,16.) 

214.  The  radius  of  a  circle  is  5(8)  (r)  ft.  Find  a  side  and 
the  area  of  the  inscribed  square. 

215.  What  is  the  radius  of  the  circle  inscribed  in  a  square 
whose  area  is  1600(5000)  (a)  sq.  ft.  ? 


118  APPLIED  MATHEMATICS 

216.  An  equilateral  triangle  is  inscribed  in  a  circle  of  radius 
6(12)  (r)  in.    Find  a  side,  the  altitude,  and  area  of  the  triangle. 

217.  The  side  of  an  inscribed  equilateral  triangle  is  9  (1.732) 
(s)  in.    Find  the  radius  of  the  circle, 

218.  The  sum  of  the  side  of  an  inscribed  equilateral  triangle 
and  the  radius  of  the  circle  is  5  +  5  Vs  (10.928)  (18)  in.  What 
is  the  length  of  a  side  and  the  radius  ? 

219.  The  area  of  a  regular  inscribed  hexagon  is  24  Vs  (17.32) 
(a)  sq.  ft.    Find  the  radius  of  the  circle. 

220.  An  equilateral  triangle  and  a  regular  hexagon  are  in- 
scribed in  a  circle.  Find  the  radius  of  the  circle  if  the  sum  of 
the  areas  of  the  triangle  and  hexagon  is  9  V3(l8V3)(389.7) 
sq.  in. 

221.  The  sum  of  the  perimeters  of  two  regular  pentagons  is 
100(225)  ft.,  and  their  areas  are  in  the  ratio  1:9(25:16). 
Find  a  side  of  each  pentagon. 

222.  The  difference  of  the  perimeters  of  two  regular  octagons 
is  40(80)  ft.,  and  their  areas  are  in  the  ratio  1 :  4(9  :  25),  Find 
a  side  of  each  octagon. 

223.  The  sum  of  the  circumferences  of  two  circles  is 
207r(176)  ft.,  and  the  difference  of  their  radii  is  2(14)  ft. 
What  are  the  radii  ? 

224.  The  radius  of  one  circle  is  6(1)  ft.  longer  than  the 
radius  of  another  circle,  and  the  sum  of  their  circumferences 
is  113f  (31.416)  ft.    Find  the  radii. 

225.  What  is  the  radius  of  a  circle  whose  area  equals  the 
area  of  two  circles  of  radii  (a)  3  and  4  in.  ?  (b)  3.3  and  5.6  cm.  ? 
(c)  6.5  and  7.2  cm.  ?  (d)  r  and  nr  ? 

226.  What  is  the  radius  of  a  circle  whose  area  equals  the 
sum  of  (a)  3,  (b)  6,  (c)  n  equal  circles  ? 

227.  What  is  the  radius  of  a  circle  that  is  doubled  in  area 
by  increasing  its  radius  1  (3)  ft.  ? 


EXERCISES  FOR  ALGEBRAIC  SOLUTION  119 

228.  A  square  and  a  circle  have  the  same  perimeter.  Find 
the  ratio  of  their  areas. 

229.  If  a  square  and  a  circle  have  the  same  area,  what  is 
the  ratio  of  their  perimeters  ? 

230.  If  a  circle  and  an  equilateral  triangle  have  the  same 
perimeter,  what  is  the  ratio  of  their  areas  ? 

231.  Construct  on  the  same  axes  curves  to  show  the  change  in 
area  of  a  circle  and  the  inscribed  regular  hexagon,  square,  and 
equilateral  triangle,  as  the  radius  increases  from  0  to  10  in. 

232.  The  area  between  two  concentric  circles  is  207r(286) 
sq.  ft.  and  the  difference  of  the  radii  is  2  (7)  ft.    Find  the  radii. 

233.  If  the  area  between  two  concentric  circles  is  96 tt  (50) 
sq.  ft.,  and  the  radius  of  the  inner  circle  is  2  (5)  ft.,  find  the 
radius  of  the  larger  circle. 

234.  In  a  circle  of  radius  12  (r)  in.  it  is  desired  to  draw  a 
concentric  circle  which  shall  bisect  the  area  of  the  given  circle. 
Find  its  radius. 

235.  The  area  of  a  circle  of  radius  8  (r)  in.  is  to  be  divided 
by  a  concentric  circle  so  that  the  area  of  the  ring  shall  be  a 
mean  proportional  between  the  area  of  the  given  circle  and  of 
the  inner  circle.   Find  the  radius. 


CHAPTER    X 
COMMON  LOGARITHMS 

68.  Definitions.  Numbers  have  been  reduced  to  powers  of 
10.    Thus  2  =  loo-^oio,  3  =  10°-*"S  125  =  lO^-^s^^. 

These  exponents  are  called  logarithms.  The  integral  part  of 
k  logarithm,  called  the  characteristic,  can  be  detennined  easily 
and  is  not  given  in  a  table  of  logarithms ;  the  decimal  par^ 
called  the  mantissa,  is  always  taken  from  the  table. 

69.  Approximate  numbers.  In  ordinary  shop  practice  and 
in  much  engineering  work  measurements  are  mad^^sually  to 
three  or  four  figures.  Thus  in  making  a  rough  estimate Hhe 
sides  of  a  building  lot  may  be  measured  to  the  nearest  foot; 
the  length  of  a  belt  may  be  measui-ed  to  the  nearest  quarter 
of  an  inch ;  an  angle  may  be  measured  to  the  nearest  tenth  of 
a  degree.  If  the  diameter  of  a  pulley  is  measured  and  said 
to  be  12.3  in.,  the  meaning  is  that  the  diameter  lies  between 
12.25  in.  and  12.35  in.,  that  is,  the  third  figure  is  doubtful. 
In  ordinary  computations,  where  numbers  with  only  three  or 
four  figures  are  involved,  a  four-place  table  of  logarithms  is 
used.  The  logarithms  are  not  exact;  they  are  approximate 
numbers  in  which  the  fourth  figure  is  doubtful.  Hence  the 
results  should  not  be  carried  beyond  four  figures. 

70.  The  mantissa.  To  find  the  mantissa  of  the  logarithm 
of  a  number  from  1  to  999,  e.g.  352,  we  look  in  the  first  column 
of  the  table  at  the  left  for  the  first  two  figures,  35,  and  in  the 
column  headed  2  we  find  the  mantissa  of  352,  namely  .5465. 
The  mantissa  of  745  is  .8722. 

(Let  the  class  read  the  mantissas  of  numbers  from  the  table 
till  all  can  find  the  mantissa  of  any  number  quickly.) 

120 


COMMON  LOGARITHMS  121 

71.  The  characteristic.    The  method  of  finding  the  charac- 
teristic is  readily  obtained  from  the  following  table  : 


10^  = 

:  1000,  . 

•.log  1000  =  3. 

log  6214 

=  3  +  a  decimal. 

10^  = 

100,    . 

•.log  100    =2. 

log  518 

=  2  +  a  decimal. 

10^  = 

10,      . 

•.log  id    =1. 

log  83 

=  1  +  a  decimal. 

10°  = 

1,    . 

•.logl        =0. 

log  6 

=  0  +  a  decimal. 

10-1  = 

.1 

•.log.l       =- 

1. 

log  .3 

=  —  1  +  a  decimal, 

10-2  = 

.01,    . 

•.log  .01     =- 

2. 

log  .04 

=  —  2  +  a  decimal. 

10-8  = 

.001,  . 

•.log  .001  =- 

3. 

log  .008 

=  —  3  +  a  decimal. 

Since  518  lies  between  100  and  1000  its  logarithm  lies  be- 
tween 2  and  3 ;  that  is,  it  is  2  plus  a  decimal. 

The  above  table  shows  that  the  characteristic  of  the  logarithm 
of  an  integer  is  one  less  than  the  number  of  integral  figures  in 
the  mimber. 

From  the  table  it  is  also  seen  that  the  characteristic  of  a 
decimal  is  a  negative  number.  Since  the  mantissa  is  always 
positive,  it  is  convenient  to  make  a  little  change  so  that  the 
characteristic  may  be  considered  positive ;  this  is  done  by 
adding  and  subtracting  10. 

Thus  log .2  =-1+  .3010  =  9.3010  -  10. 
log  .02  =  -  2  -f-  .3010  =  8.3010  -  10. 
log  .002  =  -  3  -h  .3010  =  7.3010  -  10. 

To  find  the  characteristic  of  the  logarithm  of  a  decimal,  begin 
at  the  decimal  point  and  count  the  zeros,  9,  8,  7,  •  •  •  till  the  first 
significant  figure  is  reached.  The  last  count  with  — 10  written 
after  the  mantissa  is  the  characteristic. 

72.  The  logarithm  of  a  number.  Since  10  is  the  base  of 
our  number  system,  10  is  taken  as  the  base  of  logarithms  for 
use  in  ordinary  computations.  This  makes  the  work  much 
easier,  because  the  mantissa  does  not  change  as  long  as  the 
figures  in  a  number  remain  in  the  same  order.  Thus  216,  21.6, 
.216,  and  .0216  have  the  same  mantissa. 


j^()1.8345 

=  21.6 

10 

=  10 

"1^00.8845 

=  2.16 

10 

=  10 

]^Q9.8345  - 

■10  =  .216 

J^QO.8010 

=  2. 

10« 

=  100 

122  APPLIED  MATHEMATICS 

log  216  =  2.3345,  i.e.  lO^-^^^  =  216. 

Dividing       both         10  =  10 

sides  of  the  equa-        jQTiiii  ^21.6  .'.  log 21.6  =  1.3345. 

tion  by  10,  ^^  ^  ^^ 

.log2.16  =  0.3345. 

log  .216  =  9.3345. -10. 

log  2  =  0.3010,  i.e. 

Multiplying  both 
sides  of  the  equa-        -^q2.soio        =200    .-.  log  200  =  2.3010. 
tion  by  100, 

Hence  it  is  seen  that  moving  the  decimal  point  any  number 
of  places  to  the  right  or  left  is  multiplying  or  dividing  by  some 
integral  power  of  10,  and  this  ailects  only  the  characteristic. 

The  mantissas  of  numbers  having  one,  two,  or  three  figures 
are  taken  directly  from  the  table.  The  mantissas  of  four-figure 
nmnbers  are  easily  found. 

Find  the  logarithm  of  1836.  The  mantissa  of  1836  is  the 
same  as  the  mantissa  of  183.6,  since-  moving  the  decimal  point 
does  not  change  the  mantissa.  The  mantissa  of  183.6  lies  be- 
tween the  mantissas  of  183  and  184 ;  and  it  is  .6  of  the  way 
from  the  mantissa  of  183  to  the  mantissa  of  184. 

Mantissa  of  184  -  mantissa  of  183  =  2648  -  2625 

=  23. 
23  X  .6  =  14. 
2625  +  14  =  2639. 
.-.  log  1836  =  3.2639. 
Find  log  49.23. 

Mantissa  of  493  -  mantissa  of  492  =  6928  -  6920 

=  8. 
8  X  .3  =  2. 
6920  -f-  2  =  6922. 
.-.  log  49.23  =  1.6922. 


COMMON  LOGARITHMS  123 

To  find  the  logarithm  of  a  number.  Place  the  decimal  point 
(mentally)  after  the  third  figure.  Subtract  the  next  lower  man- 
tissa from  the  next  higher.  Multiply  the  difference  by  the  fourth 
figure  of  the  number  regarded  as  tenths,  disregarding  a  fraction 
less  than  one  half  and  calling  one  half  or  more  one  ;  add  the  prod- 
uct to  the  next  lower  mantissa.    Write  the  proper  characteristic. 

(Let  the  class  find  the  logarithms  of  many  numbers.  The 
work  should  be  done  mentally ;  it  can  be  done  easily  and  quickly 
with  practice.) 

73.  To  find  a  number  from  its  logarithm.  Given  log  6  = 
1.5927,  required  to  find  b.  Looking  in  the  table  of  mantissas, 
it  is  seen  that  5927  lies  between  5922  and  5933;  the  cor- 
responding numbers  are  391  and  392.  Hence  the  nmnber  cor- 
responding to  5927  lies  between  391  and  392 ;  that  is,  it  is  391 
plus  a  fraction.  To  find  the  fraction,  add  a  zero  to  the  differ- 
ence of  the  given  mantissa  and  the  smaller,  and  divide  it  by  the 
difference  of  the  next  larger  and  next  smaller  mantissas. 

391  5922 
391.5  5927 

392  5933 

11)50(5 
Since  a  difference  of  11  in  the  mantissas  makes  a  difference  of 
1  in  the  numbers,  a  difference  of  5  makes  a  difference  of  y\  in 
the  numbers.  Hence  the  mantissa  5927  gives  the  number 
391  j\  =  391.5.  But  the  characteristic  1  shows  that  there  are 
two  integral  figures  in  the  number.    Therefore  b  =  39.15. 

Given  logm  =  0.9145,  m  =  8.213. 

log  n  =  8.8132  -  10,     n  =  .06504. 

To  find  a  number  from-  its  logarithm.  When  the  given  murv- 
tissa  lies  between  two  mantissas  in  the  table,  divide  the  differ- 
ence of  these  mantissas  into  the  difference  of  the  smaller  m-antissa 
and  the  given  mantissa,  to  one  decimal  figure.    Add  t?iis  decimal 


124  APPLIED  MATHEMATICS 

figure  to  the  number  corresponding  to  the  smaller  mantissa 
and  place  the  decimal  point  in  the  position  indicated  by  the 
characteristic. 

(All  the  work  in  finding  a  number  from  its  logarithm  should  be 
done  mentally ;  with  practice  it  can  be  done  easily  and  quickly.) 

74.  The  use  of  logarithms  in  computation.  Since  logarithms 
are  exponents  it  follows  that : 

I.  log(2x3)=log2  +  log3. 

2  =  10«-3«i»,     3  =  10o-*"i- 
2x3  =  lOO'^oio  X  10<»"^"  =  10«"»i  =  6. 
The  logarithm  of  a  product  is  equal  to  the  sum  of  the  logarithms 
of  the  factors. 

II.  log  I  =  log  3 -log  2. 

3  -J-  2  =  lO"""  ^  10»-3«"  =  10'^"«i  =  1.5. 
The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the 
dividend  minus  the  logarithm,  of  the  divisor. 

III.  log2«  =  31og2. 

28  __  /]^Q0.8010\8  ^  j^QO.9080  ^  g 

The  logarithm,  of  a  power  of  a  number  is  equal  to  the  loga- 
rithm, of  the  number  multiplied  by  the  exponent  of  the  power. 

IV.  log  V3  =  log  3*  =  i  log  3. 

V3  =  3*  =  (lO"""^)*  =  10o-238«  =  1.732. 
The  logarithm,  of  the  root  of  a  number  is  equal  to  the  logarithm 
of  the  number  divided  by  the  index  of  the  root. 

PROBLEMS 

1.  Multiply  28.34  by  3.376. 

log  28.34  =  1. 

log  3.376  =  O 

log  product  = 
product  = 


COMMON  LOGARITHMS  125 

Before  finding  the  mantissas  from  the  table  always  make  out  an 
outline  as  above.  This  saves  time  and  prevents  mistakes.  Keep  the 
signs  of  equality  and  the  figures  exactly  in  columns. 

Solution.  log  28.34  =  1.4524 

log  3.376  =  0.5284 

log  product  =  1.980^ 

product  =  95.68. 

As  a  rough  check  we  have  28^  x  3^  =  94. 

2.  Multiply  1.251  by  .6453. 

Solution.  log  1.251  =  0.0973 

log  .6453  =  9.8098  -  10 
log  product  =  9.9071  -  10 
product  =    .8074 

Rough  check.  .65  x  1|^  =    .81. 

3.  Divide  31.87  by  641.2. 

Solution.  log  31.87  =  11.5034  -  10 

log  641.2  =    2.8070 
log  quotient  =    8.6964  -  10 
quotient  =    .04970. 

Rough  check.  32  -^  640  =    .05. 

Since  the  characteristic  of  the  logarithm  of  the  divisor  is 
larger  than  the  characteristic  of  the  logarithm  of  the  dividend, 
10  is  added  to  and  subtracted  from  the  logarithm  of  the  divi- 
dend. Note  that  the  quotient  has  four  significant  figures 
(see  sect.  2).  The  zero  must  be  written  at  the  right  to  show 
that  the  division  has  been  carried  out  to  four  figures. 

4.  Divide  .8354  by  .04362. 

Solution.  log  .8354  =  9.9219  -  10 

log  .04362  =  8.6397  -  10 
log  quotient  =  1.2822 
quotient  =  19.15. 

Rough  check.  .84  ^  .044  =  19. 


126  APPLIED  MATHEMATICS 


5.  Find  .68748. 

Solution. 

log  .6874=    9.8372-10 
3 
29.5116  -  30 
log  .68748  =  9.5116 -10 
.68748  =  .3248. 

Rough  check. 

.78  =  .34. 

6.  Find  V.8231 

Solution. 

log  .8231  =  9.9155  -  10 

=  19.9155  -  20 
log  .8231  =  9.9578  -  10 
•\/.8231  =  .9074. 

Rough  check. 

Vi82  =  .9. 

Before  dividing  log  .8231  by  2,  10  was  added  and  subtracted  in 
order  that  the  resulting  logarithm  should  have  a  —  10.  Similarly, 
in  extracting  the  cube  root  of  a  decimal  add  and  subtract  20. 

7.  8.114  X  56.83.  17.  (1.237)^ 

8.  5.161  X  .0471.  18.  (.8734)^ 

9.  86.31  X  .07832.  19.   Vl983. 

10.  .0447  X  .9142.  20.   V1835. 

11.  6.320  X  3.106  X  8.141.  2I.   ^3342. 

12- mf  22.^:0687: 


13. 

^'m- 

14. 

2.178 

67.83 

15. 

.4971 

.5382 

16. 

(4.931)". 

26. 

27. 

Find  the 

112.3  ft.  wide. 

23. 
24. 
25. 


891  X  3.62  X  .5162 

68.14  X  2.657 
12.73  X  9.684 
2.056  X  .8666 ' 
4  X  3.142  X  (1.651)^ 


3 
86.3  X  4.5  X  3.142  x  15"^  x  200 
33000 
27.  Find  the  area  of  a  rectangular  lot  323.8  ft.  long  and 


COMMON  LOGARITHMS  127 

28.  The  base  of  a  triangle  is  72.14  ft.  and  its  altitude  is 
8.482  ft.   Find  its  area. 

29.  Find  the  area  of  a  square  whose  side  is  71.18  yd. 

30.  The  parallel  sides  of  a  trapezoid  are  69.14  ft.  and  38.15  ft. 
If  the  altitude  is  12.83  ft.,  find  the  area. 

31.  Find  the  surface  and  volume  of  brass  cylinders  and 
prisms,  wooden  blocks,  and  so  on. 

32.  Find  the  area  of  the  blackboard  in  square  meters. 

33.  Find  the  area  of  the  athletic  field. 

34.  Find  the  area  of  the  ground  covered  by  the  school 
buildings. 

35.  Find  the  area  of  the  block  in  which  the  school  building 
stands. 

36.  Construct  the  logarithmic  curve. 

37.  The  area  of  a  rectangle  is  1689  sq.  yd.  and  the  length 
is  58.12  yd.    Find  its  width. 

38.  Find  the  side  of  a  square  whose  area  is  77.83  sq.  ft. 

39.  The  volume  of  a  cube  is  2861  cu.  in.  Find  the  length 
of  an  edge. 

40.  What  is  the  diameter  of  a  piston  which  has  an  area  of 
172.8  sq.  in.  ? 

41.  Find  the  diameter  of  a  circular  plate  of  iron  of  the 
same  weight  and  thickness  as  a  rectangular  plate  3  ft.  4  in. 
by  2  ft.  8  in. 

42.  A  steel  shaft  is  3.5  in.  in  diameter  and  12  ft.  9  in.  long. 
Find  its  weight  if  1  cu.  in.  of  steel  weighs  .283  lb. 


CHAPTER  XI 

THE  SLIDE  RULE 

75.  Use  of  the  slide  rule.*  In  ordinary  practical  work  it  is 
usual  to  make  measurements  and  carry  results  in  computations 
only  to  three  or  four  significant  figures.  With  the  slide  rule 
multiplications  and  divisions  can  be  performed  mechanically  to 
the  degree  of  accuracy  required  in  this  work.    The  slide  rule  is 


z 

3 

4 

5   6  7  8010 

20        JO     40 

50  60  708000100 

^  1 

1 

1 

1 

Mill 

1            I         1 

1     1    1    1   11'^ 

B  1 

1 

1 

1 

1    11   II  1 

1           1        1 

1    1    1   1  1  Ib 

£ 

3 

4 
2 

5  6  78910 
3 

20       30    40 
4           5         6 

50  60108090100 
7      e     9    10 

^1 

1 

1 

1             1          1 

1       1       1      l<^ 

nl 

1 

1 

1             1          1 

1      1      1     lb 

2 

3 

4          5        6 

T     6     9   10 

Fig.  54 

widely  used  in  technical  schools  and  in  shops  and  laboratories 
where  there  is  a  large  amount  of  computation.  It  serves  as  a 
check  upon  the  numerical  solution  of  problems,  and  should  be 
used  by  engineering  students. 

76.  Description  of  the  slide  rule.  The  slide  rule  is  simply 
a  table  of  logarithms  arranged  in  such  a  way  that  they  can  be 
added  and  subtracted  conveniently.  The  logarithms  are  not 
printed  on  the  slide  rule,  but  each  number  on  it  stands  in  the 
position  indicated  by  its  logarithm.  In  Fig.  54  BC  is  the  slide, 
graduated  on  the  upper  and  on  the  lower  edges.  These  gradu- 
ations were  made  in  the  following  manner:  CC  was  divided 
into  1000  equal  parts ;  log  2  =  .301,  therefore  2  was  placed  at 

*  Cardboard  slide  rules  ready  for  the  student  to  cut  and  fit  together  may  be 
obtained  of  the  Central  Scientific  Company,  Chicago,  at  $1.10  per  dozen. 

128 


THE  SLIDE  RULE  129 

the  301st  graduation ;  log  3  =  .477,  therefore  3  was  placed  at 
the  477th  graduation ;  and  so  on  for  all  the  integers  from  1 
to  1000. 

To  read  the  numbers  from  1  to  1000  we  must  go  over  the 
rule  from  left  to  right  three  times.  Thus  we  read  first  1,  2,  3, 
•  •  • ,  10 ;  then  beginning  at  1  again  and  calling  it  10,  we  read  10, 
20,  30,  •  ■  ■ ,  100 ;  then  beginning  at  1  again  and  calling  it  100, 
we  read  100,  200,  300,  •  ■  • ,  1000. 

77.  Operations  with  the  slide  rule.  It  is  not  difficult  to 
learn  to  use  the  slide  rule  if  at  first  the  operations  are  per- 
formed with  small  numbers.  Whenever  in  doubt  about  any 
operation  perform  it  first  with  small  numbers. 

I.  Multiplication.  Multiply  3  by  2.  Move  the  slide  so  as  to 
set  1  C  on  3  D ;  then  under  2  C  read  the  product  6  on  D.  Note 
that  this  is  simply  adding  logarithms. 

To  find  the  product  of  two  numbers,  set  1  C  on  one  of  the  num- 
bers on  D,  and  tinder  the  other  number  on  C  read  the  product  on  D. 

Sometimes  in  multiplying  we  must  use  the  1  at  the  right  end 
of  scale  C  Thus  multiply  84  by  2.  Set  1  at  the  right  end  of 
scale  C  on  84  D,  under  2  C  read  168  on  D.  We  use  the  1  at 
the  left  end  or  the  right  end  of  scale  C  according  as  it  brings 
the  second  factor  over  scale  D.  In  the  example  above,  if  we 
had  set  1  at  the  left  end  of  scale  C  on  84,  then  2  C  would  have 
been  off  scale  D. 

The  decimal  point  is  placed  by  inspection.  Thus,  multiply 
12.5  by  1.8.  Set  1  C  on  18  D,  under  125  C  read  225  on  D.  But 
making  an  approximate  multiplication  mentally,  12  x  2  =  24 ; 
hence  we  know  that  there  are  two  integral  figures  in  the  prod- 
uct, giving  22.5  as  the  result.  In  all  operations  with  the  slide 
rule  the  decimal  point  can  be  placed  by  making  an  approximate 
mental  computation. 

II.  Division.  Divide  8  by  2.  Set  2  C  on  S  D,  under  1  C  read 
the  quotient  4  on  D.  Note  that  this  is  simply  subtracting 
logarithms. 


130  APPLIED  MATHEMATICS 

To  divide  one  number  by  another,  set  the  divisor  on  scale  C 
on  the  dividend  on  scale  D,  under  1  C  read  the  quotient  on  scale  D. 

The  decimal  point  is  placed  by  inspection.  Thus  divide 
3.44  by  16.  Set  16  C  on  344  D,  under  1  C  read  the  quotient 
215  on  D ;  but  we  see  that  3^16  =  about  .2 ;  hence  the  quo- 
tient is  .215. 

III.  Combined  multiplication  and  division.    Find  the  value 

24  X  9 
of  — — Set  6  C  on  24  D,  under  9  C  read  the  result  36  on 

D.  Study  this  operation  till  the  separate  parts  are  seen  clearly 
and  understood.  First  the  division  of  24  by  6  is  made  by  set- 
ting 6  C  on  24  D,  under  1  C  we  might  read  the  quotient ;  but 
we  want  to  multiply  this  quotient  by  9.  As  1  C  is  already 
on  this  quotient  we  have  only  to  read  the  product  36  on  scale 
D  under  9  C. 

An  important  problem  under  this  case  is  to  find  the  fourth 
term  of  a  proportion.    Thus,  in  the  proportion  6  :  24  =  9  :  ic, 

24  X  9 

Hence  to  find  the  fourth  term  of  a  proportion,  set  the  first 
term  on  the  second,  under  the  third  read  the  fourth. 

IV.  Continued  multiplication  and  division.  Here  for  conven- 
ience we  need  the  runner.  This  is  a  sliding  frame  carrying  a 
piece  of  glass  which  has  a  line  on  it  perpendicular  to  the  length 
of  the  rule. 

1.  Find  the  value  of  3  x  8  x  5. 

Set  1  C  at  the  right  on  3  D,  set  runner  on  8  C,  set  1  C  at  the 
right  on  the  runner,  under  5  C  read  12  on  D.    Hence 


2.  Find  the  value  of 


3x8x5  =  120. 
54 


3x6 


THE  SLIDE  RULE  131 

Set  3  C  on  54  /),  set  runner  on  1  C,  set  6  C  on  runner,  under 
1  C  read  result  3  on  D.  Note  that  we  have  simply  made  two 
divisions. 

3.  Find  the  value  of  -r- —  • 

24  X  6 

Set  24  C  on  15  D,  set  runner  on  48  C,  set  6  C  on  runner,  under 
1  C  read  result  5  on  D. 

4.  Find  the  value  of — 

Set  32  C  on  8  D,  set  runner  on  1  C,  set  1  C  at  right  end  of 
slide  on  runner,  set  runner  on  9  C,  set  1  C  on  runner,  under  4  C 
read  result  9  on  D. 

In  a  similar  manner  any  number  of  continued  multiplications 
and  divisions  may  be  performed. 

V.  Sqtiares  and  square  root.  The  graduations  on  scale  A  at 
the  top  of  the  slide  are  arranged  so  that  the  square  of  every 
number  on  scale  C  stands  directly  above  it  on  scale  A.  Thus 
above  2  is  4,  above  3  is  9,  and  above  25  is  625.  On  scale  A 
the  distances  of  the  numbers  from  1  at  the  left  end  of  the  scale 
are  proportional  to  the  logarithms  of  the  numbers  as  on  scale  C ; 
but  it  is  easier  to  learn  to  use  scale  A  by  noticing  its  relation 
to  scale  C.  We  read  from  left  to  right  1,  2,  3,  ■  •  •,  10,  20,  30, 
•  •  •,  100 ;  then  beginning  at  1  again  and  calling  it  100,  we  read 
100,  200,  300,---,  1000,  2000,  3000,  •••,  10,000.  The  first  4 
is  either  4  or  400,  that  is,  either  the  square  of  2  or  20 ;  the 
second  4  is  either  40  or  4000,  that  is,  either  the  square  of  6.32 
or  of  63.2. 

To  square  any  number,  find  the  number  on  scale  C  and  read 
its  square  directly  above  it  on  scale  A. 

To  extract  the  square  root  of  any  number,  find  the  number 
on  scale  A  and  read  its  square  root  directly  below  it  on  scale  C. 

The  upper  scale  is  very  convenient  when  multiplying  or 
dividing  by  square  roots,  finding  the  area  of  circles,  and  so  on. 


132  APPLIED  MATHEMATICS  ^ 

1.  Find  the  value  of  8  Vs. 

Set  1  C  at  right  end  of  scale  on  3  A,  under  8  C  read  result 
13.85  on  B. 

8 


2.  Find  the  value  of     ,- 
V3 


8V3 


V3         3 
Set  3  C  on  3  A,  under  8  C  read  result  4.G1  on  D. 

3.  Find  the  value  of -pz 

Set  5  5  on  8  ^,  under  12  B  read  result  4.38  on  D. 

4.  Find  the  area  of  a  circle  whose  radius  is  4  ft. 

Set  1  C  on  4  D,  above  tt  on  B  read  the  area,  50.3  sq.  ft.,  on  A. 

PROBLEMS 
1.  Find  the  value  of : 

1.  78  X  5.                  12^  ^    48^  16.8  x  4.2 

2.  38.4x25.  "15  ■  ''  2.93'  '        31.4 

3.  8.63  x  4.24.           944             84  x  13  16  V39 

4.  .121  X  6.38.           16.3 '                15      "  33 

2.  Find  the  area  of  the  rectangle  whose  dimensions   are 
3.26  in.  by  4.21  in. 

3.  The  area  of  a  rectangle  is  18.6  sq.  cm.  and  its   base  is 
5.34  cm.    Find  its  altitude. 

4.  Find  the  area  of  a  circle  whose  radius  is  (a)  5  in. ;  (b)  1.8 
in. ;  (c)  2.56  cm. ;  (d)  3.22  ft. 

5.  Construct  a  curve  to  show  the  area  of  circles  of  radius 
from  1  in.  to  10  in. 

6.  Find  the  surfaces  and  volumes  of  brass  cylinders,  prisms, 
blocks  of  wood,  and  so  on. 


THE  SLIDE  RULE  133 

7.  To  make  865  lb.  of  admiralty  metal,  used  for  parts  of 
engines  on  naval  vessels,  752.5  lb.  of  copper,  43.3  lb.  of  zinc, 
and  69.2  lb.  of  tin  were  melted  together.  Find  the  per  cent 
of  each  metal  used. 

8.  17  lb.  of  copper,  85  lb.  of  tin,  595  lb.  of  lead,  and  153  lb. 
of  antimony  were  melted  together  to  make  850  lb.  of  type 
metal.    What  per  cent  of  each  metal  was  used  ? 

9.  If  sea  water  contains  2.71  per  cent  of  salt,  how  many 
tons  of  sea  water  must  be  taken  to  give  100  lb.  of  salt  ? 

10.  The  safe  load  in  tons,  uniformly  distributed,  for  white- 
oak  beams  is  given  by  the  formula 

2hd?- 


W  = 


31 


where  W  =  the  safe  load  in  tons,  b  =  the  breadth  in  inches, 
d  =  the  depth  in  inches,  and  I  =  the  distance  between  the 
supports  in  inches. 

Construct  a  curve  to  show  the  safe  load  in  tons  for  white- 
oak  beams  having  a  breadth  of  3  in.,  distance  between  supports 
13  ft,,  and  depth  from  3  in.  to  15  in. 

11.  If  IV  =  the  weight  of  1  lb.  of  any  substance  when  sus- 
pended in  water,  and  s  its  specific  gravity,  then 

1  6-1 


1  —w 


w  = 


Construct  a  curve  showing  the  weight  of  substances  sus- 
pended in  water,  the  specific  gravity  varying  from  .5  to  15. 


CHAPTER  XII 


ANGLE  FUNCTIONS 


78.  Angles.    Let  two  lines  yiP  and  ^M  be  coincident. 

Suppose  ^P  to  revolve  about  the  point  A  away  from  AM; 
the  amount  of  turn,  indicated  by  the  arrow,  is  called  an  angle. 
The  amount  of  turn  is  expressed  in  degrees.  A  complete  turn 
gives  an  angle  of  360°,  a  half  turn  180°,  and  a  quarter  turn  90°. 
In  this  chapter  we  will  not  consider  angles  greater  than  90°. 


The  line  AM  which  marks  the  beginning  of  the  revolution 
is  called  the  initial  line;  the  line  AP  which  marks  the  ending 
of  the  revolution  is  called  the  terminal  line  of  the  angle. 

79.  Triangle  of  reference.   If  from  any  point  B  in  the 
terminal  line  of  the  angle  a  perpendicular  BC  is  dropped  to  the 
initial  line,  the  right  triangle  formed 
is  called  the  triangle  of  reference  for 
the  angle.    The  perpendicular  BC  is 
called  the  opposite  side  ;  A  C,  the  part 
of  the  initial  line  cut  off  by  the  per- 
pendicular^   is    called    the    adjacent 
side;  and  AB,  that  part  of  the  ter- 
minal line  which  belongs  to  the  triangle  of  reference,  is  called 
the  hypotenuse. 

134 


ANGLE  FUNCTIONS 


135 


80.  Sine,  cosine,  and  tangent  of 
an  angle.  Given  the  angle  A.  Con- 
struct the  triangle  of  reference,  and 
represent  the  lengths  of  the  sides  by 
a,  b,  and  c,  set  opposite  the  angles  A, 
B,  and  C  respectively. 


EC     a 
AB~  c 

_  opposite  side 
hypotenuse 

=  sin  A  (by  def- 
inition). 
This  ratio,  called  the 
sine  of  angle  yl,  is  a 
pure  number  which 
is  usually  approxi- 
mate and  expressed 
as  a  decimal. 


AC  _b 
AB~  c 

_  adjacent  side 
hypotenuse 

=  cos  A  (by  def- 
inition). 
This  ratio,  called  the 
cosine  of  angle  yl,  is 
a  pure  number  which 
is  usually  approxi- 
mate and  expressed 
as  a  decimal. 


BC^a 

AC~.b 

_  opposite  side 
adjacent  side 

=  tan  ^  (by  def- 
inition). 
This  ratio,  called  the 
tangent  of  angle  A,  is 
a  pure  number  which 
is  usually  approximate 
and  expressed  as  a 
decimal. 


These  ratios  sin  A ,  cos  A ,  and  tan  A  are  -called  functions  of 
the  angle  A  because  they  change  in  value  as  the  angle  changes. 
There  are  other  functions  of  an  angle,  but  as  these  three  seem 
to  be  the  more  important  the  discussion  will  be  limited  to  them. 


EXERCISES 

1.  Make  an  angle  A  and  construct  the  triangle  of  reference. 
Letter  as  before,  and  measure  the  sides  a,  h,  and  c  as  accurately 
as  possible  in  millimeters.  Use  the  results  of  the  measurement 
to  find  the  values  of  sin  A,  cos  A,  and  tan  A.  Carry  the  divi- 
sions as  far  as  the  errors  in  the  approximation  justify,  and 
no  farther. 

2.  Make  another  angle  A'  which  differs  from  A.  Calculate 
its  sine,  cosine,  and  tangent  in  the  same  manner.  Compare  the 
values  of  the  two  sines,  the  two  cosines,  and  the  two  tangents. 

If  you  were  to  continue  the  experiment,  you  would  find  that 
the  ratios  change  in  value  every  time  the  angle  changes  in  size. 


136 


APPLIED  MATHEMATICS 


3.  Make  an  angle  and  drop  perpendiculars  from  various 
points  on  the  terminal  line  to  the  initial  line.  Any  one  of  the 
right  triangles  may  be  considered  a  triangle  of  reference  for 
the  angle.  Find  sin  A  from  each  triangle  of  reference.  Com- 
pare the  values.  Should  they  all  be  equal  ?  Why  ?  Similarly 
for  cos  A  and  tan  A . 

4.  In  a  triangle  of  reference  ABC  could  BC  =  2  in., 
^£  =  6 in.,  and  AC  =  5m.?  Why?  Could  any  two  sides 
be  chosen  at  random  ?  Why  ?  Could 
one  side  be  chosen  at  random? 
Why? 

81.  Functions  of  45°.  Construct 
an  angle  of  45°,  and  in  the  triangle 
of  reference  make  either  AC  or  BC 
1  unit  long.  Why  is  the  other  side 
1  unit  long  ?  Why  is  the  hypotenuse 
V2  units  long  ? 


Fig.  58 


sin  45° 


1 

V2 

cos  45°=     \ 
V2 

=  iV2 

=  ^V2 

=  1(1.414) 

=  1(1.414) 

=  .707. 

=  .707. 

tan  45°  =  - 


1 
1 
=  1. 

This  ratio  is  exact. 


82.  Functions  of  30°.  Construct 
an  angle  of  30°,  and  in  the  triangle 
of  reference  make  the  side  BC  oppo- 
site 30°,  1  unit  long.  Why  is  the  hy- 
potenuse AB  2  units  long?  Why  is 
AC  V3  units  long? 

sin30°=i  cos  30°  =  -^ 

=  •'"'•  =  in  730) 

This  ratio  is  exact.  2  v  y 


=  .866. 


=  .577. 


ANGLE  FUNCTIONS 


137 


83.  Functions  of  60°.    Construct  an  angle  of  60°,  and  in 

the  triangle  of  reference  make  the  side  A  C  adjacent  to  60°, 
1  unit  long.  Why  is  the  hypotenuse  AB  2  units  long?  Why 
is  BC  VS  units  long? 

sin  60°  =  —  cos  60°  =•  i  tan  60°  =  — 

2  1 

=  1(1.732)  ^,         =.500.  =1.732. 

—  Rfifi  This  ratio  is  exact. 

Show  how  the  functions  of  60°  can  be 
found  from  the  triangle  of  reference  for  30°. 

84.  Table  of  angle  functions.   The  func- 
tions of  angles  have  been  calculated  and 
tabulated.    In  solving  problems  the  func- 
tions of  the  angle  are  taken  from  the  table.    The  functions 
of  a  few  angles,  30°,  45°,  60°,  90°,  should  be  memorized. 


Fig.  60 


PROBLEMS 


1.  A  man  standing  110  ft.  from  a  tree  on  level  ground 
finds  the  angle  of  elevation  of  the  top  of  the  tree  to  be  37°  20'. 
How  high  is  the  tree,  and  how  far  is  the  man  from  the  top  of  it  ? 


Solution.   Given 

^=37° 

20' 

h  =  110  ft. 

-  =  tanyl. 
b 

-  =  cos^. 
c 

^ 

a  =  h  tan  A 

h  =  ccos^. 

<^^/^ 

=  110  (.7627) 

h 

^^ 

=  83.9  ft. 

^  -         * 
cos^ 

110 

.7951 

=  138  ft. 

A^ 

^ 

Z- 
Fig.  61 

Check  this  and  all  problems  by  constructing  the  triangle  from  the 
given  parts.  Make  a  good-sized  drawing  to  scale  and  measure  the 
computed  parts. 


138  APPLIED  MATHEMATICS 

2.  A  railroad  track  has  a  uniform  slope  of  5°  to  the 
horizontal.    How  many  feet  does  a  train  rise  in  going  a  mile  ? 

3.  A  ladder  24  ft.  long  rests  against  a  wall.  The  foot  of 
the  ladder  is  4  ft.  4  in.  from  the  wall.  Find  the  height  of  the 
top  of  the  ladder. 

4.  The  shadow  of  a  tree  is  38  ft.  long  when  the  angle  of 
elevation  of  the  sun  is  42°.    Find  the  height  of  the  tree. 

5.  A  ship  is  sailing  northeast  12  mi.  per  hour.  How  fast  is 
she  sailing  east  ? 

6.  A  stick  8  ft.  long  stands  vertically  in  a  horizontal  plane, 
and  the  length  of  the  shadow  is  6  ft.  What  is  the  angle  of 
elevation  of  the  sun  ? 

7.  What  is  the  slope  of  a  mountain  path  if  it  rises  118  ft. 
in  a  distance  of  835  ft.  along  the  path  ? 

8.  The  top  of  a  lighthouse  is  152  ft.  above  sea  level.  If 
the  angle  of  depression  of  a  buoy  is  12°  15',  how  far  from  the 
lighthouse  is  it  ? 

9.  The  chord  of  a  circle  is  4.4  in.  and  it  subtends  at  the 
center  an  angle  of  38°.    Find  the  radius  of  the  circle. 

10.  At  a  point  212  ft.  from  the  foot  of  a  column  the  angle 
of  elevation  of  the  top  of  the  column  is  found  to  be  24°  28'. 
What  is  the  height  of  the  column  ? 

11.  A  man  6  ft.  tall  stands  4  ft.  6  in.  from  a  lamp-post.  If 
his  shadow  is  17  ft.  long,  what  is  the  height  of  the  lamp-post  ? 

12.  A  cable  is  attached  to  a  smokestack  10  ft.  below  the  top, 
and  to  a  pile  42  ft.  from  the  foot  of  the  stack.  If  the  cable 
makes  an  angle  of  62°  20'  with  the  horizontal,  find  the  height 
of  the  stack. 

13.  From  the  top  of  a  lighthouse  160  ft.  above  sea  level  two 
vessels  appear  in  line.  If  their  angles  of  depression  are  4°  20' 
and  2°  45'  respectively,  how  many  miles  are  they  apart  ? 


ANGLE  FUNCTIONS  139 

14.  As  the  angle  of  elevation  of  the  sun  increases  from 
35°  15'  to  64°  25',  how  many  feet  does  the  shadow  of  a  church 
steeple  120  ft.  high  decrease  ? 

15.  In  the  gable  shown  in  the  figure 

angie  BAF=GO°,  angle  GFE  =  30°, 

BG  =  6  ft.,  and  GE  =  4:  ft.   Find  AF, 

FE,  and  AC.  ^      „^ 

.  '  Fig.  62 

16.  The  base  AC  of  an  isosceles 

trapezoid  is  100  ft.,  and  the  equal  sides  AD  and  CB  make  angles 
of  60°  with  the  base.  The  altitude  is  40  ft.  Compute  the  length 
of  the  upper  base  and  the  area.    Draw  to  scale  and  check. 

17.  The  pitch  of  a  roof  (angle  which  the  rafters  make  with 
the  horizontal)  is  32°.  If  the  house  is  22  ft.  wide,  find  the 
length  of  the  rafters  and  the  height  of  the  gable. 

18.  A  building  80  ft.  long  and  40  ft.  wide  has  each  side 
of  its  roof  inclined  40°  to  the  horizontal.  Find  the  area  of 
the  roof. 

19.  Two  towns  A  and  B  are  at  opposite  ends  of  a  lake.  It  is 
known  that  a  station  P  is  3  mi.  from  A  and  2  mi.  from  B.  If 
the  angle  PAB  =  34°  30'  and  angle  PBA  =62°  40',  find  the 
distance  between  the  towns. 

20.  Make  a  height  or  distance  problem  of  your  own  and 
solve  it. 

85.  Logarithmic  solutions.  In  the  preceding  problems  the 
numbers  involved  consist  of  only  two  or  three  figures;  hence 
there  would  be  little  or  no  time  saved  in  using  logarithms. 
However,  when  there  are  several  figures  in  the  numbers,  and 
there  are  three  or  more  multiplications  or  divisions,  logarithms 
should  be  used. 

The  logarithms  of  the  angle  functions  are  found  in  exactly 
the  same  way  as  are  the  logarithms  of  numbers.  Thus,  find 
log  sin  18°  26'. 


140  APPLIED  MATHEMATICS 

Mantissa  log  sin  18°  30'  —  mantissa 

log  sin  18°  20'  =  5015  -  4977 
=  38. 
38  X  .6  =  23. 
4977  -f-  23  =.  5000. 
.  • .  log  sin  18°  26'  =  9.5000  -  10. 

The  sine  and  tangent  of  an  angle  increase  as  the  angle 
increases,  hence  the  difference  for  the  minutes  is  added  to  the 
mantissa  of  the  smaller  angle  taken  from  the  table. 

It  is  to  be  noted  that  the  cosine  of  an  angle  decreases  as  the 
angle  increases ;  hence  the  difference  for  the  minutes  is  to  be 
subtracted  instead  of  added. 
Thus  find  log  cos  24°  48'. 
Mantissa  log  cos  24°  40'  —  mantissa 

log  cos  24°  50'  =  9584  -  9579 
=  5. 
5  X  .8  =  4. 
9584  -  4  =  9580. 
.-.  log  cos  24°  48'  =  9.9580  -  10. 
Given  log  tan  x  —  9.5946  —  10,  find  x. 
21°  20'  5917 

21°  2  5946  . 

21°  30'  5954 

37)290(8 
.-.  a;  =  21°  28'. 

This  work  should  be  done  mentally.  In  class  find  logarithms 
of  the  functions  of  many  angles,  and  the  angles  from  the  log- 
arithms of  the  functions,  as  quickly  as  possible  until  this  can 
be  done  readily. 

The  sine  and  cosine  of  an  angle  are  always  less  than  1. 
Why  ?  Hence  the  characteristic  of  the  logarithm  is  9  — 10, 
8  —  10,  and  so  on.  The  — 10  is  not  printed  in  the  table,  but 
should  be  written  in  computation. 


ANGLE  FUNCTIONS 


141 


PROBLEMS 

1.  In  the  right  triangle  ABC, 

given                C  =  90°, 

A  =  28°  34', 

Cj/^'^ 

c  =  48.32  ft. 

A  ^ 

Find  a  and  b. 
Solution. 

FiG.  63 

a        .      . 
—  =  sinyl. 
c 

ft 

-  =  costI. 

c 

a  =  c  sin  A. 

b  =  c  cosyl. 

logc  = 
log  sin  A  = 

log 

logc  = 

cos  A  = 

loga  = 

\ogb  = 

a  = 

b  = 

B 


Before  looking  up  any  logarithms  always  make  out  an  outline 
as  above. 


-  =  Sin  A . 
c 

a  =  c  sin  A . 
log  c  =  1.6841 
log  sin  A  =  9.6796  -  10 


-  =  cos  A. 

c 

b  =  c  cosyl. 
log  c  =  1.6841 
log  cos  A  =  9.9436  -  10 
log  &  =  1.6277 
b  =  42.43  ft. 


logo  =  1.3637 
a  =  23.11  ft. 

Check.  It  may  be  as  much  work  to  check  a  problem  as  to  solve 
it,  but  an  answer  is  absolutely  worthless  unless  it  is  known  to  be 
correct.  What  is  the  advantage  of  knowing  how  to  work  problems 
if  you  cannot  get  correct  results  ? 

a2  +  b^  =  c2(Pythag.  th.). 
a2  =  c2  -  b^ 

=  (c-b)(c  +  b). 
c-b=    5.89  log  =  0.7701 
c  +  b  =  90.75  log  =  1.9579 
log  a2  =  2.7280 
log  a  =  1.3640 
a  =  23.12. 
A  difference  of  1  in  the  last  figure  may  be  expected  since  the 
logarithms  are  only  appi-oximate. 


142  APPLIED  MATHEMATICS 

2.  Two  trees  M  and  N  are  on  opposite  sides  of  a  river.  A 
line  NP  at  right  angles  to  MN  is  432.7  ft.  long  and  the  angle 
NPM  is  52°  27'.    What  is  the  distance  from  MtoN? 

3.  From  the  top  of  a  building  156.4  ft.  high  the  angle  of 
depression  of  a  street  corner  is  18°  46'.  Find  the  horizontal 
distance  from  the  street  corner  to  the  building. 

4.  To  find  the  height  of  the  Auditorium  tower  a  distance 
of  311.2  ft.  was  measured  from  the  foot  of  the  tower  and  the 
angle  of  elevation  of  the  tower  was  found  to  be  40°  57'.  Find 
the  height  of  the  tower. 

Solve  the  following  right  triangles,  two  parts  being  given : 

5.  a  =  146.8,  h  =  203.3.  9.  c  =  110.9,  a  =  64.21. 

6.  J  =  49.74,  A  =  53°  38'.  10.  b  =  8.226,  c  =  12.15. 

7.  e  =  94.53,  B  =  62°  51'.  11.  c  =  .02936,  a  =  .01153. 

8.  c  =  436.5,  A  =  74°  11'.  12.  a  =  .9681,  A  =  42°  17'. 

13.  Find  the  side  of  an  equilateral  triangle  inscribed  in  a 
circle  of  radius  52.18  in. 

14.  The  side  of  an  equilateral  triangle  inscribed  in  a  circle 
is  14.26  in.    Find  the  radius  of  the  circle. 

15.  If  a  side  of  a  regular  pentagon  is  30.24  in.,  find  the 
radius  of  the  circumscribed  circle,  and  the  apothem. 

16.  A  regular  pentagon  is  inscribed  in  a  circle  of  radius 
11.32  in.    Find  a  side  and  the  apothem  of  the  pentagon. 

17.  The  apothem  of  a  regular  polygon  of  12  sides  is  21.26  ft. 
What  is  the  perimeter  ? 

18.  The  perimeter  of  a  regular  octagonal  tower  is  168.4  ft. 
What  is  the  area  of  the  base  of  the  tower  ? 

19.  A  regular  octagonal  column  is  cut  from  a  circular  cylinder 
whose  diameter  is  18.32  in.  Find  the  area  of  a  cross  section  of 
the  column. 

20.  A  side  of  a  regular  hexagon  inscribed  in  a  circle  is 
28.43  ft.  Find  a  side  of  a  regular  decagon  inscribed  in  the 
same  circle. 


ANGLE  FUNCTIONS 


143 


r  =  sin  J.. 

0 

h  =  bsmA. 


86.  Area  of  triangles.  In  triangle 
ABC,  h  is  the  perpendicular  from 
C  toe. 

Area  triangle 

ABC  =  ^  base  x  altitude 

=  ^  eb  sin  ^ . 
The  area  of  a  triangle  eqtmls  one 
half  the  product  of  two  sides  and  the 
sine  of  the  included  angle. 

PROBLEMS 

1.  Find  the  area  of  a  triangle  ABC,  given  a  =  42.84  ft., 
c  =  76.31  ft.,  and  B  =  29°  18'. 

SoLUTioK.  2  area  =  ac  sin  B. 

log  a  =  1.6318 
log  c  =  1.8826 
log  sin  B  =  9.6896-10 
log  2  area  =  3.2040 
2  area  =  1600. 
area  =  800  sq.  ft. 

Find  the  area  of  the  following  triangles.    Check  by  finding 
the  area  twice,  using  different  angles : 

2.  a  =  34.36,  b  =  110.5,  c  =  98.32,  A  =  17°  43',  C  =  60°  36'. 

3.  a  =  88.48,  b  =  58.59,  c  =  54.38,  B  =  40°  10',  C  =  36°  47'. 

4.  a  =  1.432,  b  =  1.583,  c  =  1.610,  A  =  53°  17',  B  =  62°  24'. 

5.  a  =  3.207,  b  =  2.367,  c  =  1.435,  B  =  42°  55',  C  =  24°  22'. 

6.  Find  the  area  of  a  triangle  XYZ,  given  x  =  184.2  ft., 
ij  =  381.3  ft.,  and  Z  =  51°  24', 

7.  The  vertical  angle  of  an  isosceles  triangle  is  75°  18'  and 
the  equal  sides  are  16.46  ft.  long.  Find  the  area  of  the  triangle. 

8.  What  is  the  area  of  a  parallelogram  if  two  adjacent  sides 
are  243.6  yd.  and  315.4  yd.  and  the  included  angle  is  35°  40'  ? 


144 


APPLIED  MATHEMATICS 


9.  Two  streets  make  an  angle  of  53°  18'  with  each  other. 
The  corner  lot  between  them  has  a  frontage  of  286  ft.  on  one 
street  and  324  ft.  on  the  other.  Draw  to  scale  and  find  the  area 
of  the  lot. 

10.  Two  railroads  cross  at  an  angle  of  21°  25'.  From  a  point 
on  one  of  them  100  rd.  from  the  crossing  how  must  a  fence 
be  run  so  that  the  inclosure  shall  contain  10  A.  ? 

11.  The  survey  of  a  field  gave  the  following  data: 

EA  =  420  ft. 
EB  =  865  ft. 
EC  =  875  ft. 
ED  =  650  ft.      .  ^ 

ZAEB  =  A2°. 
ZBEC  =  36°. 
Z  CED  =  20°. 

Draw  the  field  to  scale  and  Fig.  65 

find  its  area. 

12.  A  surveyor  set  his  transit  over  the  corner  A  of  a  field 
A  BCD  and  found  the  angle  DAC  =  40°  12',  and  angle  CAB  = 
70°  54'.  AD\^  52.8  rd.,  A  C  is  86.3  rd.,  and  AB\^  38.4  rd.  Draw 
to  s(!ale  and  compute  the  area  of  the  field. 


87.  Law  of  sines.    In  the  triangle  ABC  let  h  be  the  perpen- 
dicular from  the  vertex  C  to  the  side  c. 

—  =  sin  A ,  and  —  =  sm  B. 
0  a 


By  division, 

a  _  sin  A 

a 


s,\\\B 
h 


sin  A       sin  B 


(1) 
(2) 


What  algebraic  operations  were  used  to  derive  (2)  from  (1)? 
What  theorem  in  geometry  could  be  used  for  this  purpose  ? 


ANGLE  FUNCTIONS  145 

By  dropping  a  perpendicular  from  ^  to  a  we  may  obtain,  in 
a  similar  manner, 


sin  C      sin  B 
a      _      b  G 

sin  A       sin  B      sin  C 

Law  of  Sines.  In  any  tr-ianffle  the  sides  are  proportional  to 
the  sines  of  the  opposite  angles. 

When  a  side  and  two  angles  of  a  triangle  are  given  we  may 
find  the  other  two  sides  by  this  law. 

PROBLEMS 

1.  In  a  triangle  ABC  given  A  =  36°  56',  B  =  72°  6',  and  a  = 
36.74.    Find  b  and  c. 

Solution.  C  =  180°  -  (A  +  B)  =  70°  58'. 

b  a  c  a 


sin  B      sin  A 

sin  C      sin  A 

a  sin  B 

*  =  —■ — r- 

smyl 

a  sin  C 
sin /I 

loga=    1.5652 

loga=    1.5652 

log  sin  B  =    9.9784 

-10 

log 

sin  C  =    9.9756 

-10 

11.5436 

-10 

11.5408 

-10 

log  sin  A  =    9.7788 

-10 

log 

sin  .4=    9.7788- 

-10 

logi=    1.7648 

logc=    1.7620 

b  =  58.19. 

c  =  57.81. 

Solve  the  following  triangles  and  check  by  drawing  to  scale : 

2.  A=  44°  59',  B  =  62°  52',  a  =  7.942. 

3.  A  =  50°  24',  C  =  68°  35',  b  =  12.63. 

4.  jB  =  72°46',  C  =  41°44',  c  =  203.6. 

5.  A  =  61°  18',  B  =  58°  32',  b  =  84.03. 

6.  To  find  the  distance  from  a  point  yl  to  a  point  P  across  a 
river,  a  base  line  AB  1000  ft.  long  was  measured  off  from  A. 
The  angles  BAP  and  ABP  were  found  to  be  36°  18'  and  62°  35' 
respectively.    Compute  the  distance  AP. 


146 


APPLIED  MATHEMATICS 


7.  On  board  two  ships  half  a  mile  apart  it  is  found  that  the 
angles  subtended  by  the  other  ship  and  a  fort  are  84°  16'  and 
78°  38'  respectively.  Find  the  distance  of  each  ship  from  the 
fort. 

8.  M  and  N  are  stations  on  two  hilltops  3684  ft.  apart,  and 
P  is  a  station  on  a  third  hill.  The  angles  NMP  and  MNP  are 
observed  to  be  50°  42'  and  63°  24'  respectively.  Find  the  dis- 
tances MP  and  NP. 

C 
88.  Law  of  cosines.   In  triangle  ABC, 

h  is  the  perpendicular  from  C  to  c. 

In  triangle  on  left,  i^  =  7^2  +  h'\  (1)  W 

and.  J  =  cos  J, 

or  b'  =  bc,o^A.  (2) 

In  triangle  on  right, 

=  7i2  +  c2-2i'c  + J'2. 
Substituting  (2),         =  h'' +  b'^  +  c^  -  2  be  cos  A . 
Substituting  (1),     a^  =  b^  -\-  c^  —  2bc  cos^. 
Similarly,  by  dropping  perpendiculars  from  A  and  B  we  get 

b^  =  a^  +  0^  —  2ac  cos  B. 
c^  =  a""  +  b^  -  2  ab  cos  C. 

Law  of  Cosines.  In  any  triangle  the  square  of  any  side  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides  less  twice 
the  product  of  these  two  sides  and  the  cosine  of  the  included 
angle. 

PROBLEMS 

1.  Find  a  in  the  triangle  ABC,  given  b  =  6  in.,  c  =  5  in.,  and 
A  =  29°  15'. 


Solution.  a^  =  b^  +  c^  —  2bc  cos  A 

=  36  +  25-2x6x5x 
=  8.65. 
a  =  2.9  in. 


,8725 


ANGLE  FUNCTIONS  147 

2.  In  triangle  ABC,  find  A  ii  a  =  7,  b  =  S,  e  =  9. 
Solution.  a^  =  b^  +  c^ —  2  be  coa  A. 

coaA 


i2  +  c2  - 

a2 

2  be 
64  +  81  - 

-49 

2x8x9 
=  .6667. 
A  =  48°  11'. 

3.  Find  B  and  C  in  the  triangle  in  Problem  2  and  check  by 
adding  the  three  angles. 

Solve  the  following  triangles  and  check  by  drawing  to  scale 
or  otherwise : 

4.  a  =  10,  i  =  12,  c  =  14.      6.  b  =  21,  c  =  1%  A  =  48°57'. 

5.  a  =  A,  b  =  5,  c  =  6.  7.  a  =  U,  b  =  12,  c  =  60°. 

8.  Two  ships  leave  a  dock  at  the  same  time.  One  sails  east 
12  mi.  per  hour  and  the  other  northeast  14  mi.  per  hour.  How 
far  will  they  be  apart  at  the  end  of  5  hr.  ? 

9.  From  a  point  5  mi.  from  one  end  of  a  lake  and  4  mi.  from 
the  other  end,  the  lake  subtends-  an  angle  of  56°  8'.  What  is 
the  length  of  the  lake  ? 

10.  A  and  B  are  two  stations  on  opposite  sides  of  a  moun- 
tain, and  C  is  a  station  on  top  of  the  mountain  from  which  A 
and  B  are  visible.  If  CA  =  4.2  mi.  and  CB  =  3.1  mi.,  and  angle 
ACB  =  88°  12',  find  the  distance  from  A  to  B,  the  three  stations 
being  in  the  same  vertical  plane. 

89.  Triangle  of  forces.  The  weight  W  at  the  end  of  the 
boom  is  held  in  position  by  three  forces  :  (a)  the  force  of  gravity 
acting  downward ;  (b)  the  tension  (pull)  in  the  tie ;  (c)  the 
thrust  (push)  of  the  boom.  The  tension  in  each  side  of  the 
triangle  is  proportional  to  the  lengths  of  the  sides.  The  ten- 
sion in  the  mast  is  always  taken  equal  to  the  load  W;  and  the 
tension  per  foot  is  the  same  in  each  side  of  the  triangle.  Thus  in 
Fig.  68,  ifW=  2000  lb.,  AB  =  10  ft.,  and  BC  =  16  ft.,  the  tension 
in  the  mast  AB  =  2000  lb.  and  the  tension  per  foot  =  200  lb- 


148 


APPLIED  MATHEMATICS 


Therefore  the  compression  in  the  boom  =  16  x  200  =  3200  lb. 
The  tie  A  C 


VlO'-  +  16'^  =  V356  =  18.9  ft.,  and  the  tension 
in  ^  C  ==  18.9  X  200  =  3780  lb. 

Check. 
2000^=    4,000,000 


3200'^  =  10,240,000 

14,240,000 

3780^  =  14,290,000 


Fig. 


A  Simple  Crane 


Exercise.  Put  two  screw  eyes  in  the  wall  80  cm.  apart  and 
construct  a  model  of  a  crane,  using  a  meter  stick,  string,  and  a 
spring  balance,  as  shown  in  Fig.  69.    Compute  the  tension  for 


etkm 


eocm 


Fig.  69 


different  weights  and  check  by  the  readings  of  the  spring 
balance.  After  a  weight  has  been  attached  the  string  should 
be  shortened  enough  to  make  the  string  or  the  meter  stick 
perpendicular  to  the  wall  in  order  to  form  a  right  triangle.- 


PROBLEMS 

1.  The  mast  of  a  crane  is  12  ft.  long  and  the  tie  18  ft.  The 
boom  is  horizontal  and  supports  a  load  of  2400  lb.  Find  the 
tensions  in  the  boom  and  tie. 


ANGLE  FUNCTIONS 


149 


2.  The  tie  of  a  crane  is  horizontal.  If  it  is  24  ft.  long  and 
the  boom  is  30  ft.  long,  find  the  tension  in  the  mast,  boom,  and 
tie  for  a  load  of  4  T. 

3.  The  tie  of  a  crane  makes  an  angle  of  30°  with  the  mast, 
and  the  boom  is  horizontal.  If  the  boom  is  20  ft.  long  and  the 
load  is  3000  lb.,  find  the  tension  in  the  mast,  tie,  and  boom. 

4.  The  boom  of  a  crane  is  16  ft.  long  and  makes  an  angle 
of  40°  with  the  liiast.  The  tie  is  horizontal.  Find  the  tension 
in  the  mast,  boom,  and  tie  for  a  load  of  2  T. 

5.  The  boom  of  a  crane  is  20  ft.  long,  and  when  it  is  hori- 
zontal the  tie  is  30  ft.  long.  If  the  tie  can  stand  a  strain  of 
4200  lb.,  find  the  greatest  load  that  can 

be  lifted  when  the  boom  is  horizontal. 

6.  The  bracket  BCD  carries  a  load  of 
4001b.  at  D.  Find  the  stresses  in  BC, 
CD,  and  BD. 

7.  An  arc  lamp  weighing  20  lb.  is 
hung  on  a  pole,  as  shown  in  Fig.  71. 
Find  the  stresses  in  MP  and  NP. 

8.  A  weight  of  96  lb.  is  attached  to  a 
cord  which  is  secured  to  the  wall  at  a  point 
A  and  is  pushed  out  from  the  wall  by  a 
horizontal  stick  BC.  If  .4C  =  6ft.  and 
angle  BAC  =  38°,  find  the  tension  in  AB 
and  the  pressure  on.  BC. 

9.  A  canal  boat  is  kept  20  ft.  from  the 
towpath  and  the  towline  is  72  ft.  long.  If 
there  is  a  pull  of  144  lb.  on  the  line,  what 
is  the  effective  pull  ? 


Fig.  70 


VI 


/v 


Fig.  72 


Solution.    Let  C,  Fig.  72,  be  the  position  of  the  canal  boat. 
AB=  V722  _  202  =  69.2  ft. 
iy^-  =  2  1  b.,  the  tension  i^er  foot  in  A  C. 
.-.  69.2  X  2  =  138.4  lb.,  the  effective  pull. 


150 


APPLIED  MATHEMATICS 


10.  The  pull  on  the  towline  of  a  canal  boat  is  400  lb.  and 
the  line  makes  an  angle  of  10°  with  the  direction  of  the  boat. 
How  much  of  the  pull  is  effective  ?  How  much  is  at  right 
angles  to  the  direction  of  the  boat  ? 

11.  A  boat  is  pulled  up  the  middle  of  a  stream  60  ft.  wide 
by  two  men  on  opposite  sides,  each  pulling  with  a  force  of 
100  lb.  If  each  rope,  attached  to  the  bow  of  the  boat,  is  40  ft. 
long,  find  the  effective  pull  on  the  boat. 

12.  Each  of  two  horses  attached  to  a  load  is  pulling  with  a 
force  of  200  lb.  If  they  are  pulling  at  an  angle  of  60°  with 
each  other,  what  is  the  effective  pull  on  the  load  ? 

13.  Attach  two  spring  balances  to  the  wall,  as  shown  in 

Fig.  73,  with  10  or  12  ft.  of  cord  be-    _^ a b. 

tween  them.  At  the  center  of  the  cord 
attach  an  8-lb.  weight.  Kead  each  bal- 
ance for  the  tension  in  ^  C  and  BC. 

Suppose  AC  =  6  ft.  and  DC  =  4  ft. 
Compute  the  stress  in  AC. 
Solution. 


Fig.  73 


1  of  8  =  4  lb.,  stress  in  DC. 

I  =  lib.  per  foot,  stress  in  DC. 
1x6  =  6  lb.,  stress  in  A  C. 

Compare  with  result  of  the  experiment.  Make  other  experi- 
ments with  different  lengths  of  cord  until  the  reason  for  the 
method  of  computation  is  understood,     /t  \      s'    C        s' 

14.  A  man  weighing  180  lb.  sits  in 
the  center  of  a  hammock  12  ft.  long. 
If  the  supports  are  10  ft.  apart,  find 
the  pull  on  the  hammock. 

Solution. 


\  of  180 


CD  =  V62  -  52  =  3.32  ft. 
=  90  lb.,  pull  in  CD. 

=  pull  per  foot. 


90 
3.32 
90  X  6 

3.32 
163  lb.  =  pull  on  hammock. 


=  163  lb.,  pull  in  AD. 


ANGLE  FUNCTIONS  161 


Check.  cos  a;  =  ^ 

=  .8333. 

X  =  33°  34'. 

Pull  in  CD       . 

=  sm  X. 

Full  in  AD 

Pull  in  CD  =  sin  a:  X  pull  in  AD 

=  .553  X  163 

=  90.2  lb. 

90.2  X  2  =  180.4  lb.,  weight  of  the  man. 

15.  Two  horses  attached  to  a  load  are  pulling  with  the  same 
force  at  an  angle  of  60°  with  each  other.  If  the  combined 
effective  pull  on  the  load  is  400  lb.,  how  many  pounds  is  each 
horse  pulling  ? 

16.  Connect  two  light  strips  of  wood  60  cm.  long,  AB  and  BC 
(Fig.  75),  by  a  hinge  at  B,  and  put  casters  at  A  and  C.  Put  a 
cord  and  spring  balance  between  A  b 

and  C,  as  shown  in  the  figure.  Hold  yi^^^^^f^^^^^ 

the  frame  vertical,  measure  BD  and  y^     j       n:;^ 

AC,  and   read   the   balance,  when     ^^  j  ^^ 

BD  =  48  cm.  and  AD  =  36  cm.    At-   ^^^^       I  U 

tach  an  8-lb.  weight  at  B  and  make  Fig.  75 

AC  =  72  cm.  Read  the  balance,  and 

subtract  the  first  reading  to  get  the  tension  in  ^C  due  to  the 
8-lb.  weight.    Compute  the  tension  in  ^  C  as  follows  : 

•  4*^  =  y^Tj  lb.,  tension  per  centimeter  in  BD. 
jV  X  36  =  3  lb.,  tension  in  AD. 
.'.  tension  in  ^C  =  3  lb. 

Compare  with  the  result  of  the  experiment.  Make  other  ex- 
periments with  different  weights  and  distances  AC,  until  the 
reason  for  the  method  of  computation  is  understood. 

17.  A  pair  of  rafters  supports  a  weight  equivalent  to  800  lb. 
at  the  ridge.  The  pitch  of  the  roof  is  30°  and  the  width  of 
the  building  is  30  ft.  Find  the  tension  in  the  tie  through  the 
foot  of  each  rafter. 


152 


APPLIED  MATHEMATICS 


18.  The  width  of  a  house  is  24  ft.  and  the  rafters  are  16  ft. 
long.  If  the  rafters  support  a  weight  equal  to  600  lb.  at  the 
ridge,  find  the  stress  in  the  rafters. 

19.  A  bridge  truss  ABC  supports 
a  weight  of  300  lb.  per  foot  horizon- 
tally. The  span  is  30  ft.  long.  If 
CD  =  10  ft.,  find  the  stresses  in  A  C 
and  AB.  (The  load  at  D  equals  one 
half  the  total  load.) 

20.  ABC  (Fig.  77)  is  an. inverted 
king-post  truss.  AB  =  20  ft.,  and  the 
angles  CAB  and  ABC  =  40°.  If  the 
load  at  Z)  is  4  T.,  find  the  stresses  in 
A  C  and  AB. 


CHAPTER  XIII 

GEOMETRICAL  EXERCISES  FOR  ADVANCED  ALGEBRA 

90.  A  figure  should  be  drawn  for  each  exercise,  letters  or 
numbers  put  on  the  lines  in  the  figure,  and  the  equations  set 
up  from  the  figure.  Check  by  drawing  to  scale  and  measuring 
the  required  parts.  The  first  exercises  involve  square  roots, 
since  radicals  are  reviewed  early.  Some  of  the  exercises  should 
be  worked  out  in  notebooks,  with  emphasis  placed  on  accuracy 
in  drawing  and  neatness  in  arrangement. 

1.  Construct  a  graph  for  the  squares  of  numbers  from  0  to 
13.  Units  :  horizontal,  1  large  square  =  1 ;  vertical,  1  large 
square  =  10.  What  is  the  equation  of  the  curve  ?  Find  V 2, 
Ve,  Vt^,  Va25,  VlO,  VI2,  and  Vl2!5  to  three  decimal 
places  and  check  by  the  graph. 

2.  Find  the  diagonal  of  a  square  whose  side  is  12  (.s). 


Fig.  78 


Solution.    (1"^=  12^  +  122(Pythag.  th.)  iP  =  s^  +  .s2(Pvtha,o:.  th.) 
=  288  '  =2s2 

f/  =  V288  d  =  .sV2. 

=  Vl44  X  2 
=  12V2 

=  16.97.  • 

153 


154  APPLIED  MATHEMATICS 

3.  Find  the  side  of  a  square  whose  diagonal  is  5  (d). 

4.  The  side  of  an  equilateral  triangle  is  4(s).    Find  the 
altitude  and  area. 

5.  The  altitude  of  an  equilateral  triangle  is  6(A).    Find  the 
side  and  area. 

6.  The  area  of  an  equilateral  triangle  is  24(a).    Find  the 
side  and  altitude. 

7.  Find  the  area  of  a  regular  hexagon  whose  side  is  3(s). 

8.  Find  the  area  of  a  regular  hexagon  whose  apotheni  is 
2  (A). 

9.  Find  the  side  and  apothem  of  a  regular  hexagon  whose 
area  is  36  (a). 

10.  A  star-shaped  figure  is  formed  by  constructing  equilateral 
triangles  outwardly  on  the  four  sides  of  a  square.  If  the  area 
of  the  entire  figure  is  100,  find  a  side  of  the  square, 

11.  Squares  are  constructed  outwardly  on  the  sides  of  a 
regular  hexagon.  If  the  area  of  the  entire  figure  is  72,  find  a 
side  of  the  hexagon. 

12.  From  a  square  whose  side  is  12  (s)  a  regular  octagon  is 
formed  by  cutting  off  the  corners.    Find  a  side  of  the  octagon. 

13.  The  edge  of  a  cube  is  5  (le).    Find  a  diagonal. 

14.  The  diagonal  of  a  cube  is  8  (cT).   Find  an  edge. 

15.  Find  the  diagonal  of  a  rectangular  parallelepiped  whose 
edges  are  4,  5,  and  6  (a,  b,  and  c). 

16.  Find  the  side  of  an  equilateral  triangle  whose  area  equals 
the  area  of  a  square  whose  diagonal  is  6  V50. 

17.  Two  sides  of  a  triangle  are  a  and  b.  Show  that  the  area 
is  \  ab  when  the  included  angle  is  30°  or  150°. 

18.  If  two  sides  of  a  triangle  are  a  and  b,  show  that  the  area 
is  1  V3  ab  when  the  included  angle  is  60°  or  120°. 

19.  If  two  sides  of  a  triangle  are  a  and  b,  show  that  the  area 
is  I  V2  ab  when  the  ijicluded.  angle  is  45°  or  136°. 


GEOMETRICAL  EXERCISES  155 

20.  The  sides  of  a  triangle  are  30,  60,  and  80  (a,  b,  and  c). 
Find  the  segments  of  each  side  formed  by  the  bisector  of  the 
opposite  angle. 

21.  The  shadow  cast  upon  level  ground  by  a  certain  church 
steeple  is  27  (37)  yd.  long,  and  at  the  same  time  the  shadow  of 
a  vertical  rod  5  (7)  ft.  high  is  3  (6)  ft.  long.  Find  the  height 
of  the  steeple. 

22.  The  footpaths  on  the  opposite  sides  of  a  street  are  30  ft. 
apart.  On  one  of  them  a  bicycle  rider  is  moving  at  the  rate 
of  15  mi.  per  hour.    If  a  man  on  the  other  side,  walking  in  the 

.  opposite  direction,  regulates  his  pace  so  that  a  tree  5  ft.  from 
his  path  continually  hides  him  from  the  rider,  at  what  rate 
does  he  walk  ? 

23.  One  side  of  a  triangle  is  divided  into  two  equal  parts 
and  through  the  point  of  division  a  line  is  drawn  parallel  to 
the  base.  Into  what  fractional  parts  is  the  triangle  divided  ? 
Similarly,  when  the  side  is  divided  into  3,  4,  5,  •  •  • ,  n  equal  parts  ? 

24.  Find  the  side  of  an  equilateral  triangle  if  the  center  of 
gravity  is  2  (a;)  in.  from  the  vertex. 

25.  What  part  of  a  triangle  lies  between  the  base  and  a  line 
through  the  center  of  gravity  parallel  to  the  base  ? 

26.  One  side  of  a  triangle  is  10  (s)  in.  Where  must  a  point 
be  taken  in  the  given  side  in  order  that  a  line  drawn  through 
it,  parallel  to  another  side,  will  divide  the  triangle  into  two 
areas  whose  ratio  is  3  :  4  (m  :  w)  ? 

27.  The  bases  of  a  trapezoid  are  16  and  10 (b^  and  b^  and  the 
altitude  is  6(A).  Find  the  area  of  the  triangle  formed  by 
producing  the  nonparallel  sides  of  the  trapezoid. 

28.  Find  the  side  of  the  square  inscribed  in  the  triangle 
whose  base  is  12(b)  and  altitude  is  6(h). 

29.  A  rectangle  whose  length  is  twice  its  breadth  is  inscribed 
in  an  equilateral  triangle.  Find  the  area  of  the  rectangle  if  a 
side  of  the  triangle  is  2. 


156  APPLIED  MATHEMATICS 

30.  Find  the  area  of  a  trapezoid,  given  the  bases  36  and  56 
(b^  and  h^  and  the  altitude  12  (A). 

31.  The  bases  of  a  trapezoid  are  73  and  57  (/'j  and  h^  and 
each  of  the  nonparallel  sides  is  17  (c).    Find  the  area. 

32.  One  diagonal  of  a  trapezoid  is  10((/).  The  segments  of 
the  other  diagonal  are  6  and  9  (m  and  n).  Find  the  segments 
of  the  first  diagonal. 

33.  A  trapezoid  contains  480  (65)  sq.  ft.  and  its  altitude  is 
20  (10)  ft.  Find  the  bases  of  the  trapezoid  if  one  of  them  'is 
4  (6)  ft.  longer  than  the  other. 

34.  Find  the  area  of  a  rectangle  if  its  diagonal  is  50  (il)  ft. 
and  the  sides  are  in  the  ratio  3  :  5  (m  :  n). 

35.  The  dimensions  of  a  rectangle  are  64  and  58  (i  and  h) 
respectively.  If  the  length  is  diminished  by  10  (m),  how  much 
must  the  breadth  be  increased  in  order  to  retain  the  same  area  ? 

36.  A  rectangle  is  8  (Ji)  in  breadth  and  its  diagonal  is  20  (d). 
Upon  the  diagonal  as  a  base  a  triangle  is  constructed  whose 
area  is  equal  to  that  of  the  rectangle.  Find  the  altitude  of  the 
triangle. 

37.  The  ratio  of  the  diagonals  of  a  rhombus  is  7  :  5(m  :  n) 
and  their  sum  is  16  (Jc).    Find  the  area  of  the  rhombus. 

38.  The  sides  of  a  right  triangle  are  x,  x  -\-l,  and  a;  +  8. 
Find  them. 

39.  Two  telegraph  poles  25  and  30  ft.  high  are  80  ft.  apart 
on  level  ground.    Find  the  length  of  the  wire. 

40.  The  chord  of  a  circle  is  8(c)  and  the  height  of  the  seg- 
ment is  2  (A).    Find  the  radius. 

41.  In  a  circle  whose  radius  is  12(r)in.  a  chord  4(c)  in.  is 
drawn.    Find  the  height  of  the  segment. 

42.  Two  chords  48  and  14  mm.  long  are  on  opposite  sides  of 
the  center  of  a  circle.  If  they  are  31  mm.  apart,  what  is  the 
diameter  of  the  circle  ? 


GEOMETRICAL  EXERCISES  157 

43.  Two  parallel  chords  on  the  same  side  of  the  center  of  a 
circle  are  48  and  14  (14  and  4)  in.  long.  If  the  diameter  of 
the  circle  is  50(16)  in.,  find  the  distance  between  the  chords. 

44.  Find  the  common  chord  of  two  equal  circles  of  radius 
8  (r)  in.  if  each  circle  has  its  center  on  the  circumference  of 
the  other. 

45.  Two  chords  AB  and  CD  intersect  at  E  within  a  circle. 
IiAE  =  10(4),  BE  =  12  (9),  and  CD  =  23  (12),  find  CE  and  ED. 

46.  From  a  point  without  a  circle  a  secant  and  a  tangent  are 
drawn.  If  the  external  segment  of  the  secant  is  6  (m)  and  the 
internal  segment  is  18  (n),  find  the  tangent. 

47.  If  from  a  point  without  a  circle  two  secants  are  drawn 
whose  external  segments  are  3  and  4(3.2  and  5.5),  and  the  in- 
ternal segment  of  the  latter  is  17(4.7),  what  is  the  internal 
segment  of  the  former  ? 

48.  The  distance  between  the  centers  of  two  circles  whose 
radii  are  5  and  8(r^  and  r^)  is  26(d).  How  far  from  the  center 
of  each  circle  does  their  common  tangent  intersect  the  line  of 
centers  ?    (Two  solutions.) 

49.  The  radii  of  two  circles  are  7  in.  and  4  in.  The  distance 
between  their  centers  is  12  in.  Find  the  length  of  the  common 
internal  and  external  tangents. 

50.  Find  the  radius  of  a  circle  if  tlie  numerical  measure  of 
the  area  equals  the  measure  (a)  of  the  radius ;  (Jj)  of  the 
circumference. 

51.  Find  the  side  of  the  largest  square  piece  of  timber  that 
can  be  cut  from  a  log  14  ft.  in  circumference. 

52.  A  rectangle  and  a  circle  have  equal  perimeters.  Find  the 
difference  of  their  areas  if  the  radius  of  the  circle  is  9  (12)  in. 
and  the  width  of  the  rectangle  is  ^  (J)  its  length. 

53.  A  circle  whose  radius  is  8(?-)  has  one  half  of  its  area 
removed  by  cutting  a  ring  from  the  outside.  What  is  the  width 
of  the  ring  ? 


158  APPLIED  MATHEMATICS 

54.  Show  that  the  ratio  of  the  square  inscribed  in  a  semi- 
circle to  the  square  inscribed  in  the  entire  circle  is  2  :  5. 

55.  Show  that  the  ratio  of  the  square  inscribed  in  a  semi- 
circle to  the  square  inscribed  in  a  quadrant  of  the  same  circle 
is  8  :  5. 

56.  What  is  the  ratio  of  the  square  inscribed  in  a  quadrant 
of  a  circle  to  the  square  inscribed  in  the  entire  circle  ? 

57.  How  much  must  be  added  to  the  circumference  of  a  wheel 
whose  radius  is  2  (r)  to  make  the  radius  1  (m)  longer  ? 

58.  If  an  electric  cable  were  laid  around  the  earth  at  the 
equator,  how  many  feet  would  have  to  be  added  if  the  cable 
were  raised  10  ft.  above  the  surface  of  the  earth  ? 

59.  A  quarter-mile  running  track  is  to  be  laid  out  with  straight 
parallel  sides  and  semicircular  ends.  The  track  is  to  be  10  ft. 
wide,  and  the  distance  between  the  outer  parallel  edges  is  to 
be  220  ft.  What  must  be  the  extreme  length  of  the  field  so 
that  a  runner  may  cover  the  exact  quarter  of  a  mile  by  keeping 
in  the  center  of  the  track  ? 

60.  In  any  triangle  whose  sides  are  a,  b,  and  c  derive  a 
formula  for  the  square  of  the  side  opposite  an  acute  angle. 

61.  Derive  a  corresponding  formula  for  the  square  of  the 
side  opposite  an  obtuse  angle. 

62.  In  a  triangle  whose  sides  are  7,  8,  and  9(4,  5,  and  6) 
find  the  projections  of  the  sides  7  and  8  (4  and  5)  on  9  (6). 

63.  In  a  triangle  whose  sides  are  10,  12,  and  18  (40,  80,  and 
100)  find  the  projections  of  the  sides  18  and  10(100  and  40) 
on  12(80). 

64.  The  sides  of  a  triangle  are  4,  5,  and  7(70,  90,  and  100). 
Find  the  altitude  to  base  7  (100). 

65.  Find  the  area  of  a  triangle  whose  sides  are  8, 12,  and  15 
(20,  25,  and  30). 


GEOMETRICAL  EXERCISES  159 

66.  Find  the  length  of  the  common  chord  of  two  circles 
whose  radii  are  5  and  8  (10  and  17)  and  the  distance  between 
whose  centers  is  10(21). 

67.  The  base  and  altitude  of  a  triangle  are  8  and  6  (b  and  h)  in. 
respectively.  If  the  base  be  increased  4  (c)  in.,  how  much  must  the 
altitude  be  diminished  in  order  that  the  area  remain  the  same  ? 

68.  Through  the  vertex  of  a  triangle  whose  area  is  120 
(100)  sq.  in.  a  line  is  drawn  dividing  it  into  two  parts,  one 
containing  24  (12)  sq.  in.  more  than  the  other.  What  are  the 
segments  into  which  the  base  is  divided  if  the  whole  base  is 
20 (14)  in.? 

69.  The  base  of  a  triangle  is  6  in.  and  the  altitude  is  5  in. 
Find  the  change  in  area  if  the  dimensions  are  (a)  increased  by 
3  in.  and  2  in.  respectively  ;  (b)  diminished  by  3  in.  and  2  in. 
respectively ;  (c)  one  increased  by  3  in.  and  the  other  dimin- 
ished by  2  in.    What  is  the  per  cent  of  change  in  each  case  ? 

70.  At  a  distance  of  60  ft.  from  a  building  the  angles  of 
elevation  of  the  top  and  bottom  of  a  tower  on  the  building  are 
45°  and  30°  respectively.    Find  the  height  of  the  tower. 

71.  If  the  shadow  of  a  tree  is  lengthened  60  (a)  ft.  as  the 
angle  of  elevation  of  the  sun  changes  from  45°  to  30°,  how 
high  is  the  tree  ? 

72.  A  ladder  resting  against  a  vertical  wall  forms  an  angle 
of  60°  with  the  level  ground.  If  the  foot  of  the  ladder  is  drawn 
10  ft.  farther  out  from  the  wall,  the  angle  formed  with  the 
ground  is  30°.    Find  the  length  of  the  ladder. 

73.  In  a  right  triangle  whose  legs  are  12  and  16(20  and  40) 
find  the  length  of  the  perpendicular  from  the  vertex  of  the  right 
angle  to  the  hypotenuse,  and  the  segments  of  the  hypotenuse. 

74.  The  legs  of  a  right  triangle  are  9  and  12  (a  and  b).  Find 
their  projections  on  the  hypotenuse. 

75.  The  projections  of  the  legs  of  a  right  triangle  on  the 
hypotenuse  are  5f  and  9f  (m  and  n).    Find  the  legs. 


160  APPLIED  MATHEMATICS 

76.  The  sum  of  the  three  sides  of  a  right  triangle  is  60  (140)  in. 
and  the  hypotenuse  is  26(58)  in.  Find  the  legs  and  the  per- 
pendicular from  the  vertex  of  the  right  angle  to  the  hypotenuse. 

77.  In  a  right  triangle  the  perpendicular  from  the  vertex  of 
the  right  angle  to  the  hypotenuse  is  2  (^>)  and  the  ratio  of  the 
segments  of  the  hypotenuse  is  4::9(m:n).  Find  the  area  of 
the  triangle. 

78.  The  perpendicular  from  the  vertex  of  the  right  angle 
in  a  triangle  makes  the  segment  of  the  hypotenuse  adjacent  to 
the  longer  leg  equal  to  the  shorter  leg.  Find  the  area  of  the 
triangle  when  the  hypotenuse  is  2(c). 

79.  Two  roads  cross  at  right  angles  at  ^.  5  mi.  from  A  on 
one  road  a  man  travels  toward  A  at  the  rate  of  3  mi.  per  hour. 
6  mi.  from  A  on  the  other  road  another  man  travels  toward  A 
at  the  rate  of  6  mi.  per  hour.  When  and  where  will  the  men 
be  2  mi.  apart  ? 

80.  Two  trains  run  at  right  angles  to  each  other,  one  at  30 
and  the  other  at  40  mi.  per  hour.  The  first  train  is  15  mi.  from 
the  crossing  and  is  moving  away  from  it ;  the  second  is  60  mi. 
from  the  crossing  and  moving  toward  it.  When  and  where 
will  the  trains  be  50  mi.  apart  ? 

81.  How  much  must  the  length  of  a  rectangle  16  by  12 
(b  by  h)  be  increased  in  order  to  increase  the  diagonal  4  (c)  ? 

82.  The  difference  between  the  diagonal  of  a  square  and 
one  of  its  ^ides  is  2.071  (a)  in.    Find  one  side  and  the  area. 

83.  Find  the  sides  of  a  rectangle  if  the  perimeter  is  34  (p)  in. 
and  the  diagonal  is  13  (cT)  in. 

84.  The  diagonal  and  longer  side  of  a  rectangle  are  together 
5  times  the  shorter  side,  and  the  longer  side  exceeds  the  shorter 
by  7.    What  is  the  area  of  the  rectangle  ? 

85.  The  perimeter  of  a  right  triangle  is  24(216)  and  the 
area  is  24(1944).  Find  the  sides.  (Solve  with  one,  then  with 
two,  and  then  with  three  unknowns.) 


GEOMETRICAL  EXERCISES  161 

86.  From  a  square  piece  of  tin  a  box  is  formed  by  cutting 
6-in.  squares  from  the  corners  and  folding  up  the  edges.  If 
the  volume  of  the  box  is  864  (1944)  cu.  in.,  what  was  the  size 
of  the  original  piece  of  tin  ? 

87.  The  sum  of  the  volumes  of  two  cubes  is  35  (2728)  cu.  in. 
and  the  sum  of  an  edge  of  each  is  5  (22)  in.    Find  their  diagonals. 

88.  If  the  edges  of  a  rectangular  box  were  increased  by  2,  3, 
and  4  in.  respectively,  the  box  would  become  a  cube  and  its 
volume  would  be  increased  by  1008  cu.  in.  Find  the  edges  of 
the  box. 

89.  The  diagonal  of  a  box  is  125  in.,  the  area  of  the  lid  is 
4500  sq.  in.,  and  the  sum  of  the  three  coterminous  edges  is 
215  in.    Find  the  three  dimensions. 

90.  A  rectangular  piece  of  cloth  shrinks  5  per  cent  in  length 
and  2  per  cent  in  width.  The  shrinkage  of  the  perimeter  is 
38  in.  and  of  the  area  862.5  sq.  in.  Find  the  dimensions  of 
the  cloth. 

91.  If  a  given  square  be  subdivided  into  four  (n^^  equal 
squares  and  a  circle  inscribed  in  each  of  these  squares,  the 
sum  of  the  areas  of  these  circles  will  equal  the  area  of  the 
circle  inscribed  in  the  original  square. 

92.  In  a  square  whose  side  is  16  a  square  is  inscribed  by 
joining  the  mid-points  of  the  sides  in  order.  In  this  square 
another  square  is  inscribed  in  a  similar  manner.  This  is  re- 
peated indefinitely.  Find  the  area  of  the  first  eight  inscribed 
squares. 

93.  In  any  triangle  a  triangle  is  inscribed  by  joining  the 
mid-points  of  the  sides.  Another  triangle  is  inscribed  in  this 
inscribed  triangle  in  a  similar  manner,  and  so  on  indefinitely. 
How  does  the  area  of  the  sixth  triangle  compare  with  the  area 
of  the  first  ? 

94.  An  equilateral  triangle  is  circumscribed  about  a  circle 
of  radius  4(r).    Find  a  side  of  the  triangle.  • 


162  APPLIED  MATHEMATICS 

95.  A  circle  is  inscribed  in  a  triangle  whose  sides  are  5,  6, 
and  7  (a,  h,  and  c).  Find  the  distances  of  the  points  of  contact 
from  the  vertices  of  the  triangles. 

96.  Find  the  radius  of  the  circle  inscribed  in  an  isosceles 
trapezoid  whose  bases  are  6  and  18  (l)^  and  h.^. 

97.  A  boy  places  his  eyes  at  the  surface  of  a  smooth  body 
of  water  and  finds  that  the  top  of  a  float         f. 
1  mi.  away  is  just  visible.    How  far  does 
the  float  project  above  the  water  ? 

Solution,  a;  (8000  +  a;)  =  1^. 

x^  +  8000  x  =  \. 

Since  x  is  very  small  compared  with  the 

diameter  of  the  earth,  we  may  drop  x^. 

5280  X  12  . 

X  = m.  Fig.  79 

8000 

98.  A  man  6  ft.  tall  standing  on  the  seashore  sees  an  object 
on  the  horizon.  How  far,  in  miles,  is  the  object  away  from 
the  shore  ? 

99.  From  the  top  of  a  cliff  60  ft.  high  is  barely  visible  the 
funnel  of  a  steamer,  known  to  be  30  ft.  above  the  surface.  How 
far  is  the  steamer  from  the  cliff  ? 

100.  The  bridge  of  a  steamer  is  40  ft.  above  the  water.  How 
far  apart  are  two  such  steamers  when  the  bridge  of  one  is  just 
visible  from  the  bridge  of  the  other  ? 

101.  In  a  circle  whose  radius  is  5  (r)  a  chord  8  (c)  is  drawn. 
Find  the  length  of  the  chord  of  one  half  the  arc. 

102.  Find  the  side  of  a  regular  polygon  of  twelve  sides  in- 
scribed in  a  circle  of  radius  6  (r). 

103.  Find  the  side  of  a  regular  octagon  inscribed  in  a  circle 
of  radius  8  (r). 

104.  The  area  inclosed  by  two  concentric  circles  is  60  (a)  sq.  ft. 
If  the  radius  of  the  inner  circle  is  5  (r)  ft.,  find  the  radius  of 
the  outer  circle. 


GEOMETRICAL  EXERCISES  163 

105.  Three  men  buy  a  grindstone.  If  the  diameter  is  3  (d)  ft., 
how  much  of  the  radius  must  each  man  grind  off  in  order  to 
obtain  his  share  ? 

106.  The  sum  of  the  circumferences  of  two  circles  is  56^  ft. 
and  the  sum  of  their  areas  is  141f  sq.  ft.    Find  their  radii. 

(TT  =  V-.) 

107.  The  area  of  a  rectangular  table  whose  length  is  5  ft. 
more  than  its  breadth  is  equal  to  the  area  of  a  circular  table 
whose  radius  is  3^  ft.    Find  the  dimensions  of  the  table. 

108.  On  a  straight  line  8  (w)  cm.  long  as  a  diameter  describe 
a  semicircle.  On  each  half  of  the  given  line  as  diameters  de- 
scribe semicircles  within  the  other  semicircle.  Find  the  radius 
of  the  circle  which  is  tangent  to  the  three  semicircles. 

109.  An  increase  of  2  ft.  in  one  side  of  an  equilateral  triangle 
enlarges  the  area  by  4  VS  sq.  ft.    Find  the  side  of  the  triangle. 

110.  The  sides  of  a  rectangle  are  8  and  12  (b  and  h).  Find 
the  area  of  an  equilateral  triangle  whose  sides  pass  through 
the  vertices  of  the  rectangle. 

111.  The  number  which  expresses  the  area  of  a  right  triangle 
is  1  greater  than  the  number  which  expresses  the  length  of 
the  hypotenuse.  Show  that  the  sum  of  the  legs  of  the  triangle 
is  2  greater  than  the  hypotenuse. 

112.  Find  the  side  of  the  square  inscribed  in  the  common 
part  of  two  circles  of  radius  6  (/■),  if  the  center  of  each  circle  is 
on  the  circumference  of  the  other. 

113.  Two  parallel  lines  are  8  and  12  in.  long  respectively, 
and  are  4  in.  apart.  Find  the  area  of  the  two  triangles  formed 
by  joining  their  opposite  extremities. 

114.  How  many  squares  may  be  inscribed  in  a  triangle 
whose  sides  are  9,  12,  and  15? 

115.  In  a  triangle  whose  sides  are  3,  3,  and  4  (a,  a,  and  c)  a 
line  drawn  across  the  sides  3  and  4  (b  and  c)  bisects  both  the 
perimeter  and  the  area.  How  far  from  the  vertex  does  the  line 
cut  the  sides  ? 


CHAPTER  XIV 

VARIATION 

91.  Direct  variation.  If  a  man  earns  |25  per  week,  the 
amount  he  earns  in  a  given  time  equals  |25  multiplied  by  the 
number  of  weeks. 

a  =  25  n. 


Number  of  weeks , 
Amount  earned     . 


1 
25 


2 
50 


4 

100 


5 
125 


As  the  number  of  weeks  changes  the  amount  earned  changes, 
but  always  the  amount  earned  divided  by  the  immber  of  weeks 
equals  25. 

^  =  25. 


We  may  state  this  fact  in  another  way  and  say  that  the  amount 
earned  varies  directly  as  the  number  of  weeks,  or  a  cc  n. 

If  a  steel  rail  weighs  100  lb.  per  yard,  the  weight  of  the  rail 
equals  the  length  in  yards  multiplied  by  100.    w  =  100  I,  or 

w 

—  =  100.    Since  the  weight  divided  by  the  length  is  constant, 

100,  we  may  state  this  fact  in  the  form  of  variation,  and  say 
that  the  weight  varies  directly  as  the  length,  ov  w  ccl. 

Note  that  in  direct  variation  an  increase  in  one  variable 
makes  an  increase  in  the  other.  The  greater  the  length  the 
greater  the  weight;  the  less  the  length  the  less  the  weight. 
Double  the  length  and  the  weight  is  doubled ;  one  fourth  of 
the  length  gives  one  fourth  the  weight. 

164 


VARIATION  165 

92.  Definition.  One  niunber  varies  directly  as  another  when 
the  quotient  of  the  first  divided  by  the  second  is  constant. 

Exercise.  On  a  sheet  of  squared  paper  take  the  lines  at  the 
bottom  and  left  for  the  axes  of  x  and  y  respectively,  and  let 
one  square  each  way  equal  one.  Draw  a  straight  line  from  the 
lower  left  corner  to  the  intersection  of  any  two  heavy  lines. 

Make  a  table  for  the  values  of  x,  y,  and  -  for  points  on  this 

line,  taking  a;  =  1,  2,  3,  •  •  •,  10.  Is  the  quotient  of  y  divided 
by  X  constant  ?  Does  y  vary  directly  as  ic  ?  What  equation 
connects  y  and  x  ? 

PROBLEMS 

1.  The  weight  of  a  mass  of  brass  varies  directly  as  its 
volume.  If  150  cu.  in.  weigh  45  lb.,  how  many  cubic  inches 
weigh  7.5  lb.  ? 

Solution.    Given  wocr.  (1) 

w 
By  definition,  -  =  k.  (2) 

w  =  kv.  (3) 

Substitute  values,  45  =  ^  •  150.  (4) 

Solving  for  k,  k  =  .3.  (5) 

Substitute  in  (3),  7.5  =  .3  v.  (6) 

V  =  25  cu.  in. 

Arithmetical  solution. 

45 
The  weight  of  1  cu.  in.=  -— -  =  .3  lb. 

150 

7  5 
Hence  it  requires  -^  =  25  cu.  in.  to  weigh  7.5  lb. 

.o 

2.  Construct  a  graph  to  show  the  relation  between  the  vol- 
ume and  weight  in  Problem  1.  What  is  the  equation  of  the 
straight  line  ?  Bead  off  some  sets  of  values  from  the  graph 
and  check  by  the  equation. 

3.  The  weight  of  a  mass  of  gold  varies  directly  as  its  vol- 
ume.   If  60  cu.  in.  weighs  42  lb.,  find  the  weight  of  35  cu.  in. 


166  •         APPLIED  MATHEMATICS 

4.  Construct  a  graph  to  show  the  relation  between  the  vol- 
ume and  weight  of  a  mass  of  gold  on  the  same  axes  as  in 
Problem  2.  What  does  the  difference  in  the  slope  of  the  two 
graphs  show  ? 

5.  The  distance  through  which  a  body  falls  from  rest  varies 
as  the  square  of  the  time  during  which  it  falls.  If  a  body  falls 
400  ft.  in  5  sec,  how  far  will  it  fall  in  20  sec.  ? 

Suggestion,  d  cct^.  Check  by  arithmetical  solution.  20  -^  .5  =  4. 
Since  the  distance  varies  as  the  square  of  the  time,  the  body  will  fall 
400  X  42  =  6400  ft.  in  20  sec. 

6.  Construct  a  graph  to  show  the  relation  between  distance 
and  time  in  the  case  of  a  falling  body. 

93.  Inverse  variation.    A  man  wishes  to  lay  out  a  flower 

bed  containing  120  sq.  ft.   If  he  makes  it  12  ft.  long,  it  must  be 

10  ft.  wide ;  20  ft.  long,  6  ft.  wide  ;  and  so  on.   The  greater  the 

length  the  less  the  width.    If  the  length  is  doubled,  the  width 

120 
is  halved ;  always  lb  =  120,  or  ^  =  -y—  •    We  say  that  the  length 

.      .          1 
varies  inversely  as  the  width,  and  write  it  ^  oc  -  • 

94.  Definition.  One  number  varies  inversely  as  another 
when  their  product  equals  a  constant. 

Exercise  1.  Suspend  a  meter  stick  at  its  center  so  as  to  bal- 
ance, and  attach  a  500-g.  weight  6  cm.  from  the  fulcrum.  Sus- 
pend on  the  other  side  a  100-g.  weight  to  balance.  How  far 
from  the  fulcrum  is  it  ?  Suspend  other  weights  to  balance,  and 
make  a  table  for  the  weights  and  distances  from  the  fulcrum. 
Multiply  each  weight  by  its  distance  from  the  fulcrum.  What 
seems  to  be  true  ?    If  w -d  =  3000  (a  constant),  we  may  say 

that  the  distance  varies  inversely  as  the  weight,  dec  —  - 

w 

Exercise  2.  Locate  on  squared  paper  the  points  from  the  table 
in  Exercise  1,  and  draw  a  curve  through  them.  Express  the 
relation  between  x  and  y  (1)  as  a  variation ;  (2)  as  an  equation. 


VARIATION  167 

PROBLEMS 

1.  The  time  it  takes  to  do  some  work  varies  inversely  as  the 
number  of  men  at  work.  If  6  men  can  do  the  work  in  10  da., 
how  long  will  it  take  5  men  to  do  it  ? 

Solution.   Let  t  =  number  of  days. 

n  =  number  of  men. 

Given  to:--  (1) 

n 

By  definition,                                                 nt  =  k.  (2) 

Substitute  the  given  values  in  (2),     6  x  10  =  t.  (3) 

1  =  60.  -                 (4) 

Substitute  in  (2),                                        5  i  =  60.  (5) 

t  =  12.  (6) 

Check  by  aritlimetic.  If  6  men  can  do  the  work  in  10  da.,  1  man 
can  do  it  in  60  da. ;  and  5  men  in  ^  of  60  =  12  da. 

2.  The  number  of  hours  in  a  railway  journey  varies  inversely 
as  the  speed.  If  it  takes  7  hr.  to  go  from  Chicago  to  St.  Louis 
at  40  mi.  per  hour,  how  long  would  it  take  at  50  mi.  per  hour  ? 

3.  The  weight  of  a  body  varies  inversely  as  the  square  of 
its  distance  from  the  center  of  the  earth.  If  a  man  weighs 
200  lb.  on  the  surface  gf  the  earth  (4000  mi.  from  the  center), 
how  much  will  he  weigh  when  he  is  in  a  balloon  6  mi.  from 
the  surface  ? 

95.  Joint  variation.  If  a  carpenter  saws  a  2-in.  plank  into 
strips  of  various  lengths  and  widths,  the  volume  of  each  strip 
equals  twice  the  length  by  the  width,  or  v  =  2U).  We  may  say 
that  the  volume  vai'les  jointly  as  the  length  and  width,  and  write 
it  in  the  form  v  oc  U). 

The  number  of  cubic  feet  in  a  rectangular  water  tank  8  ft. 
high  varies  jointly  as  the  length  and  width,  since  the  number 
of  cubic  feet  =  8  Ih. 

96.  Definition.  One  number  varies  jointly  as  two  others 
when  the  first  varies  as  the  product  of  the  other  two. 


168  APPLIED  MATHEMATICS 

PROBLEMS 

1.  The  volume  of  a  cylinder  varies  jointly  as  the  altitude 
and  the  square  of  the  radius  of  the  base.  When  the  altitude 
is  20  in.  and  the  radius  of  the  base  is  10  in.,  the  volume  is 
6284  cu.  in.  Find  the  volume  when  the  altitude  is  8  in.  and 
the  radius  of  the  base  is  6  in. 

Solution.  v  x  hr^. 

6284  =  yt.20. 102. 
k  =  3.142. 
V  =  3.142  X  8  X  62 
=  904.9  cu.  in. 

2.  The  pressure  of  wind  on  a  plane  surface  varies  jointly  as 
the  area  of  the  surface  and  the  square  of  the  velocity  of  the 
wind.  If  the  pressure  on  100  sq.  ft.  is  125  lb.  when  the  wind 
is  blowing  16  mi.  per  hour,  what  will  be  the  pressure  on  a  plate- 
glass  window  10  by  12  ft.  when  the  velocity  of  the  wind  is 
70  mi.  per  hour  ? 

97.  Suggestions  for  the  solution  of  problems  in  variation. 

1.  From  the  conditions  given  in  the  problem  write  the 
variation. 

2.  Change  the  variation  to  an  equation. 

3.  Substitute  the  given  numbers  and  find  the  value  of  the 
constant  k. 

4.  In  the  equation  substitute  the  value  of  k  and  the  other 
numbers  given  in  the  problem. 

5.  Solve  this  equation  for  the  required  number. 

6.  Check. 

While  most  of  the  problems  in  the  following  list  should  be 
solved  by  the  principles  of  variation,  some  of  them  may  be 
solved  more  easily  by  proportion.  All  results  should  be  checked, 
and  as  far  as  possible  the  meaning  of  the  constant  should  be 
discussed. 


VARIATION  169 

PROBLEMS 

1.  The  circumference  of  a  circle  varies  directly  as  its  diam- 
eter, and  when  the  diameter  is  17.5  in.  the  circumference  is 
55.0  in.    Find  the  circumference  when  the  diameter  is  22.7  in. 

2.  The  velocity  acquired  by  a  falling  body  varies  as  the 
time  of  falling.  If  the  velocity  acquired  in  4  sec.  is  128.8  ft. 
per  second,  what  velocity  will  be  gained  in  7  sec.  ? 

3.  The  weight  of  a  mass  of  gold  varies  directly  as  its  vol- 
ume. If  5  ccm.  weighs  96.3  g.,  how  many  cubic  centimeters  will 
weigh  1kg.  ? 

4.  The  area  of  the  surface  of  a  cube  varies  directly  as  the 
square  of  its  edge.  What  will  be  the  edge  of  a  cube  the  area 
of  whose  surface  is  315|  sq.  in.,  if  the  area  of  the  surface  of 
a  cube  whose  edge  is  3  J  in.  is  73 1-  sq.  in.  ? 

5.  The  simple  interest  on  a  sum  of  money  varies  as  the 
time  during  which  it  bears  interest.  If  the  interest  on  a  certain 
sum  is  |84.20  for  5  yr.,  what  will  be  the  interest  for  8^  yr.  ? 

6.  The  safe  working  load  on  a  rope  varies  as  the  square  of 
its  girth.  If  the  safe  load  on  a  manila  rope  6  in.  in  girth  is 
1.2  T.,  find  the  girth  of  a  rope  whose  safe  load  is  3.6  T. 

7.  If  the  friction  between  a  wagon  and  the  roadway  varies 
as  the  total  load  on  the  wheels,  and  if  the  friction  is  24  lb. 
when  the  load  is  650  lb.,  find  the  friction  when  the  load  is  1^  T. 

8.  The  distance  a  body  falls  under  the  action  of  gravity 
varies  as  the  square  of  the  time  of  falling.  If  a  body  falls 
403  ft.  in  5  sec,  in  how  many  seconds  will  it  fall  680  ft.  ? 

9.  The  surface  of  a  sphere  varies  as  the  square  of  its  radius. 
If  the  surface  of  a  sphere  is  616  sq.  in.,  by  how  much  must  its 
radius  of  7  in.  be  increased  in  order  to  double  its  surface  ? 

10.  Given  that  the  extension  of  a  spring  varies  as  the  stretch- 
ing force,  and  that  a  spring  is  stretched  10  in.  by  a  weight  of 
5.2  lb.,  what  weight  will  stretch  the  spring  7.5  in.  ? 


170  APPLIED  MATHEMATICS 

11.  The  safe  load  on  a  rectangular  beam  varies  jointly  as 
the  breadth  and  the  square  of  the  depth.  If  a  2  by  4  in.  pine 
joist  of  given  length  supports  safely  320  lb.,  what  weight  will 
a  2^  by  10  in.  beam  of  the  same  material  and  length  safely 
support  ? 

12.  The  weight  of  a  disk  of  copper  cut  from  a  sheet  of  uni- 
form thickness  varies  as  the  square  of  the  radius.  Find  the 
weight  of  a  circular  piece  of  copper  12  in.  in  diameter,  if  one 
7  in.  in  diameter  weighs  4.42  oz. 

13.  The  volume  of  a  quantity  of  gas  varies  as  the  absolute 
temperature  when  the  pressure  is  constant.  If  a  quantity  of 
gas  occupies  3.25  cu.  ft.  when  the  temperature  is  14°  C,  what 
will  be  its  volume  at  56.5°  C.  ? 

(Absolute  temperature  =  273°  +  the  reading  of  the  Centigrade 
thermometer.) 

14.  If  the  volume  of  a  certain  gas  is  376  ccm.  when  the 
temperature  is  12°  C,  at  what  temperature  will  the  volume  be 
633.3  ccm.,  the  pressure  remaining  the  same  ? 

15.  Find  the  volume  of  a  gas  at  —  23°  C,  if  its  volume  is 
200  ccm.  at  27°  C. 

16.  If  the  quantity  of  water  that  flows  through  a  circular 
pipe  varies  as  the  square  of  the  diameter  of  the  pipe,  and  if 
1.02  gal.  per  minute  flow  through  a  half-inch  pipe,  how  many 
gallons  per  minute  will  flow  through  a  3-in.  pipe  ? 

17.  The  safe  load  on  a  wrought-iron  chain  varies  as  the 
square  of  the  diameter  of  the  section  of  the  metal  forming  a 
link.  If  the  safe  load  on  a  chain  in  which  the  metal  is  §  in. 
thick  is  900  lb.,  what  diameter  of  metal  will  be  necessary  in  a 
chain  that  is  to  bear  a  load  of  6.4  T.  ? 

18.  What  is  the  safe  load  for  a  chain  in  which  the  diameter 
of  a  section  of  the  metal  forming  a  link  is  .9  in.  ? 

19.  The  quantity  of  heat  generated  by  an  electric  current  in 
a  given  conductor  for  a  given  time  varies  as  the  square  of  the 


VARIATION  171 

number  of  amperes.  Find  the  amount  of  heat  generated  by  a 
current  of  25  amperes,  if  224  units  of  heat  are  generated  by 
a  current  of  16  amperes. 

20.  A  current  is  found  to  generate  350  units  of  heat  in  the ' 
conductor  of  J?roblem  19.    How  many  amperes  in  the  current  ? 

21.  The  compression  of  a  spring  under  a  given  load  varies 
as  the  cube  of  the  mean  diameter  of  the  coils,  other  conditions 
being  the  same.  When  the  diameter  is  4  in.  the  compression 
is  1.64  in.  What  is  the  compression  when  the  diameter  is  6^  in.  ? 

22.  The  deflection  by  a  given  load  at  the  middle  of  a  beam 
supported  at  both  ends  varies  as  the  cube  of  its  length.  A 
beam  9  ft.  long  is  deflected  .135  in.  by  a  certain  load.  Find  the 
deflection  of  a  beam  15  ft.  long  by  the  same  load. 

23.  The  diagonal  of  a  cube  varies  directly  as  the  edge  of 
the  cube.  If  the  diagonal  of  a  cube  is  8.66  in.  when  its  edge 
is  5  in.,  what  will  be  the  edge  of  a  cube  whose  diagonal  is 
13.4  in.  ? 

24.  A  solid  sphere  of  radius  3.5  in.  weighs  12  lb.  What  is 
the  diameter  of  a  sphere  of  the  same  material  that  weighs 
96  lb.,  given  that  the  weight  of  a  sphere  varies  as  the  cube  of 
its  radius  ? 

25.  The  distance  in  miles  of  the  offing  at  sea  varies  as  the 
square  root  of  the  height  in  feet  of  the  eye  above  the  sea  level. 
If  the  distance  is  4  mi.  when  the  height  is  10  ft.  8  in.,  find  the 
distance  when  the  height  is  121.5  ft. 

26.  According  to  Boyle's  law  the  volume  of  a  gas  varies 
inversely  as  the  pressure  when  the  temperature  is  constant. 
If  the  volume  of  a  gas  is  600  ccm.  when  the  pressure  is  60  g. 
per  square  centimeter,  find  the  pressure  when  the  volume  is 
150  ccm. 

27.  If  the  volume  of  a  gas  is  42.5  cu.  in.  at  a  pressure  of 

12.6  lb.  per  square  inch,  find  the  pressui-e  when  the  volume  is 

35.7  cu.  in. 


172  APPLIED  MATHEMATICS 

28.  The  pressure  allowed  in  a  cylindrical  boiler  varies  in- 
versely as  its  diameter.  When  the  diameter  is  42  in.  the 
pressure  allowed  is  104  lb.  per  square  inch.  What  pressure  is 
allowed  when  the  diameter  is  96  in.  ? 

29.  Equal  quantities  of  air  are  on  opposite  sides  of  a  piston  of 
a  cylinder  16  in.  long.  If  the  piston  moves  4  in.  from  the  center, 
find  the  ratio  of  the  pressures  on  the  two  sides  of  the  piston. 

30.  The  intensity  of  light  varies  inversely  as  the  square  of 
the  distance  from  the  source  of  light.  If  the  illumination  of  a 
gas  jet  at  a  distance  of  10  ft.  is  /,  what  will  it  be  at  20  ft.  ? 
at  50  ft.  ? 

31.  A  student  lamp  and  a  gas  jet  illuminate  a  screen  equally 
when  it  is  placed  12  ft.  from  the  former  and  20  ft.  from  the 
latter.    Compare  the  relative  intensities  of  the  two  lights. 

32.  How  far  from  a  lamp  is  a  point  that  receives  three  times 
as  much  light  as  another  point  20  ft.  away  ? 

33.  How  much  farther  from  a  gas  jet  must  a  book,  which  is 
18  in.  away  from  it,  be  removed  in  order  that  it  may  receive 
two  thirds  as  much  light  ? 

34.  An  8  candle  power  electric  lamp  at  a  distance  of  6  ft. 
from  a  screen  illuminates  it  with  one  half  the  intensity  of  a 
candle  at  a  distance  of  1  ft.  6  in.  from  the  screen.  What  is  the 
candle  power  of  the  candle  ? 

35.  In  a  given  latitude  the  time  of  vibration  of  a  pendulum 
varies  as  the  square  root  of  its  length.  If  a  pendulum  39.1  in. 
long  vibrates  once  in  a  second,  what  is  the  length  of  a  pendu- 
lum that  vibrates  twice  in  a  second  ?  three  times  in  a  second  ? 

36.  The  velocity  with  which  a  liquid  flows  from  an  orifice 
varies  as  the  square  root  of  the  head  (depth  of  the  liquid  above 
the  orifice).  A  reservoir  40  ft.  high  is  filled  with  water,  and 
when  an  opening  is  made  in  the  side  at  a  height  of  4  ft.,  the 
water  escapes  with  an  initial  velocity  of  48  ft.  per  second. 
What  would  be  the  velocity  if  the  opening  were  made  at  a 
height  of  8  ft.  ? 


VARIATION  1T3 

37.  The  weight  of  a  body  varies  inversely  as  the  square  of 
its  distance  from  the  center  of  the  earth.  If  a  body  weighs 
100  lb.  at  the  earth's  surface  (4000  mi.  from  the  center),  what 
would  be  its  weight  at  the  summit  of  the  highest  mountain, 
which  is  5^  mi.  high  ? 

38.  How  far  above  the  earth's  surface  must  a  body  that 
weighs  150  lb.  at  the  surface  be  removed,  in  order  that  its 
weight  may  be  reduced  to  96  lb.  ? 

39.  The  diameter  of  the  rivets  used  for  a  plate  varies  as  the 
square  root  of  its  thickness.  If  li-in.  rivets  are  used  for  a  1-in, 
plate,  what  size  of  rivets  is  required  for  a  |-in.  plate  ?  What 
thickness  of  plate  can  be  riveted  with  |-in.  rivets  ? 

40.  The  volume  of  a  gas  varies  inversely  as  the  height  of 
the  mercury  in  a  barometer,  the  temperature  being  constant. 
If  a  certain  mass  occupies  32  cu.  in.  when  the  barometer  reads 
28.8  in.,  what  space  will  it  occupy  when  the  reading  is  30.4  in.  ? 

41.  Compare  the  amounts  of  heat  received  at  two  points 
whose  distances  from  the  source  of  heat  are  in  the  ratio  4 : 3, 
assuming  that  the  intensity  of  heat  varies  inversely  as  the 
square  of  the  distance  from  the  source  of  heat. 

42.  If  the  attraction  of  a  magnet  for  a  piece  of  iron  varies 
inversely  as  the  square  of  the  distance  between  them,  and  if 
the  attraction  at  the  distance  of  .1  in.  is  a,  what  will  be  the 
attraction  at  .2  in.  ?  at  .3  in.  ?  at  .5  in.  ? 

43.  The  attractive  force  between  two  oppositely  electrified 
balls  varies  inversely  as  the  square  of  the  distance  between 
them.  At  a  distance  of  8  cm.  the  force  is  3.5  g.  At  what 
distance  will  the  force  be  .64  g.  ? 

44.  The  compression  of  a  spring  under  a  given  load  varies 
inversely  as  the  fourth  power  of  the  diameter  of  a  cross  section 
of  the  steel  in  the  coils,  other  conditions  being  the  same.  If 
the  compression  is  3.5  in.  when  the  diameter  is  |  in.,  what  will 
be  the  compression  when  the  diameter  is  1;^  in.  ? 


174  APPLIED  MATHEMATICS 

45.  If  7  men  in  9  weeks  earn  $516.60,  how  many  men  will 
it  take  to  earn  |360.80  in  4  weeks,  it  being  given  that  the 
amount  earned  varies  jointly  as  the  number  of  men  and  the 
number  of  weeks  ? 

46.  The  volume  of  a  circular  disk  varies  jointly  as  its  thick- 
ness and  the  square  of  its  radius.  Two  metallic  disks  having 
thicknesses  5  cm.  and  3  cm.,  and  radii  12  cm.  and  20  cm. 
respectively,  are  melted  and  recast  into  a  single  disk  6  cm. 
thick.    What  is  its  radius  ? 

47.  The  weight  of  a  metal  cylinder  varies  jointly  as  its  length 
and  the  square  of  its  diameter.  If  a  cylinder  12  in.  long  and 
4|  in.  in  diameter  weighs  49  lb.,  what  is  the  diameter  of  a 
cylinder  20  in.  long  that  weighs  135  lb.  ? 

48.  The  volume  of  a  cone  varies  jointly  as  its  altitude  and 
the  square  of  the  radius  of  its  base.  If  the  volume  of  a  cone 
is  4.95  ccm.  when  its  altitude  is  2.1  cm.  and  its  radius  is  1.5  cm., 
find  the  altitude  of  a  cone  whose  radius  is  3  cm.  and  volume 
33  ccm. 

49.  How  far  from  a  light  of  9  candle  power  will  the  illumi- 
nation be  2^  times  the  illumination  at  a  distance  of  24  ft.  from 
a  light  of  16  candle  power  ? 

50.  The  weight  of  a  uniform  bar  of  given  material  varies 
jointly  as  its  length  and  the  area  of  its  cross  section.  If  a  steel 
bar  1  sq.  in.  in  cross  section  and  1  ft.  long  weighs  3.3  lb.,  what  is 
the  weight  of  a  T-rail  2  ft.  long  and  8|  in.  in  cross-sectional  area  ? 

51.  An  ohm  is  the  resistance  offered  to  the  flow  of  an  electric 
current  through  a  column  of  mercury  106  cm.  long  and  1  sq.  mm. 
in  cross-sectional  area.  What  is  the  resistance  of  a  column  of 
mercury  3  m.  long  and  4  sq.  mm.  in  cross-sectional  area,  the 
resistance  varying  directly  as  the  length  and  inversely  as  the 
cross-sectional  area  ? 

52.  A  wire  of  diameter  .0704  in.  has  a  resistance  of  15  ohms. 
Find  the  diameter  of  a  wire  of  the  same  length  and  material 
whose  resistance  is  5.4  ohms. 


VARIATION  176 

53.  If  the  resistance  of  500  yd.  of  a  certain  cable  is  .65  ohm, 
what  will  be  the  resistance  of  1  mi.  of  a  cable  of  the  same 
material  and  of  one  half  the  cross-sectional  area  ? 

54.  Find  the  resistance  of  1000  yd.  of  copper  wire  .16  in.  in 
diameter,  if  the  resistance  of  112  yd.  of  copper  wire  .06  in.  in 
diameter  is  1  ohm.  Solve  also  by  the  formula  in  Exercise  10, 
page  73,  and  compare  the  results.  See  Chapter  XVII  for  defi- 
nitions and  explanations. 

55.  The  resistance  of  a  certain  wire  is  1.82  ohms,  and  the 
resistance  of  2^  mi.  of  the  same  wire  is  known  to  be  3.25  ohms. 
Find  the  length  of  the  first  wire. 

56.  The  resistance  of  2400  ft.  of  a  certain  copper  wire  of 
cross  section  11.2  sq.  mm.  is  1.13  ohms.  What  is  the  resistance 
of  2  mi.  of  copper  wire  of  cross  section  6.45  sq.  mm.  ? 

57.  According  to  Ohm's  law  the  number  of  amperes  flowing 
through  an  electric  circuit  varies  directly  as  the  number  of  volts 
of  electromotive  force  and  inversely  as  the  number  of  ohms 
resistance.  If  the  voltage  in  a  certain  circuit  is  such  as  to  main- 
tain a  current  of  10  amperes  through  a  resistance  of  40  ohms, 
what  would  be  the  current  if  the  electromotive  force  were 
doubled  and  the  resistance  diminished  by  one  third  ? 

58.  How  many  amperes  are  there  in  the  current  maintained 
by  a  dynamo  whose  resistance  is  2.4  ohms,  that  of  the  rest  of 
the  circuit  being  17.6' ohms,  and  the  electromotive  force  210 
volts  ? 

59.  The  resistance  offered  by  the  air  to  the  passage  of  a 
bullet  through  it  varies  jointly  as  the  square  of  its  diameter 
and  the  square  of  its  velocity.  If  the  resistance  to  a  bullet 
whose  diameter  is  .32  in.  and  whose  velocity  is  1562.5  ft.  per 
second  is  67.5  oz.,  what  will  be  the  resistance  to  a  bullet  whose 
diameter  is  .5  in.  and  whose  velocity  is  1300  ft.  per  second  ? 

60.  From  the  data  of  Problem  59  determine  the  diameter 
of  a  bullet  that  has  a  resistance  of  50  oz.  when  its  velocity  is 
900  ft.  per  second. 


176  APPLIED  MATHEMATICS 

61.  If  t  denotes  the  time  of  revolution  of  a  planet  in  its  orbit 
about  the  sun,  and  d  the  mean  distance  of  the  planet  from  the 
sun,  then  f'  varies  as  d^.  Assuming  that  the  earth's  period  of 
revolution  is  365  da.  and  that  of  Venus  225  da.,  find  the  ratio 
of  the  mean  distances  of  these  two  planets  from  the  sun. 

62.  The  horse  power  that  a  solid  steel  shaft  can  transmit 
safely  varies  jointly  as  its  speed  in  revolutions  per  minute 
and  the  cube  of  its  diameter.  A  4-in.  solid  steel  shaft  making 
120  r.  p.  m.  can  transmit  240  h.p.  How  many  horse  power  can 
be  transmitted  if  the  diameter  of  the  shaft  is  3  in.  and  its  speed 
100  r.  p.  m.  ? 

63.  The  pressure  of  the  wind  on  a  plane  surface  varies 
jointly  as  the  area  of  the  surface  and  the  square  of  the  wind's 
velocity.  If  the  pressure  on  a  square  yard  is  12^  lb.  when  the 
velocity  of  the  wind  is  17^  mi.  per  hour,  what  is  the  pressure 
on  a  square  foot  when  the  velocity  of  the  wind  is  45  mi.  per 
hour  ? 

64.  The  space  s  passed  over  and  the  time  of  flight  ^  of  a 
body  projected  vertically  upward  are  connected  by  the  rela- 
tion s  =  at  —  16 1"^,  where  a  is  constant.  If  s  =  676  ft,  when 
t  =  Q\  sec,  find  s  when  i  =  3  sec. 


CHAPTER  XV 

EXERCISES  IN  SOLID  GEOMETRY 

98.  Use  short  methods  of  multiplication  and  division  and 
keep  the  results  to  a  reasonable  number  of  significant  figures. 

I.  Numerical  Exercises 

1.  A  line  8  ft.  long  makes  with  a  plane  an  angle  of  45°. 
Find  the  length  of  the  projection  of  the  line  upon  the  plane. 

2.  What  will  be  the  length  of  the  projection  of  the  line 
in  the  preceding  exercise,  if  it  makes  an  angle  of  30°  with 
the  plane  ? 

3.  Prove  that,  if  a  line  is  inclined  to  a  plane  at  an  angle  of 
60°,  its  projection  upon  the  plane  is  equal  to  half  the  line. 

4.  In  a  swimming  tank  the  water  is  5^  ft.  deep  and  the 
ceiling  is  11  ft.  above  the  water ;  a  pole  22  ft.  long  rests 
obliquely  on  the  bottom  of  the  tank  and  touches  the  ceiling. 
How  much  of  the  pole  is  above  the  water  ? 

5.  From  a  point  P  6  in.  from  a  plane  a  perpendicular  PQ 
is  drawn  to  the  plane ;  with  Q  as  a  center  and  a  radius  of 
^\  in.  a  circle  is  described  in  the  plane ;  at  any  point  R  of 
this  circle  a  tangent  RT  10  in.  long  is  drawn.  Find  ihe  dis- 
tance from  P  to  T". 

6.  With  a  12-ft.  pole  marked  in  feet  how  can  you  deter- 
mine the  foot  of  the  perpendicular  let  fall  to  the  floor  from 
the  ceiling  of  a  room  9  ft.  high  ? 

7.  If  a  point  is  20  cm.  from  each  of  the  vertices  of  a  right 
triangle  whose  legs  are  12  cm.  and  16  cm.  respectively,  find 
the  distance  from  the  point  to  the  plane  of  the  triangle. 

177 


178  APPLIED  MATHEMATICS 

8.  Determine  the  relation  between  (a)  the  edge  and  the 
diagonal  of  a  face  of  a  cube ;  (b)  the  edge  and  a  diagonal  of 
a  cube, 

9.  The  sum  of  the  squares  of  the  three  edges  of  a  rectangular 
parallelepiped  is  2166  and  the  three  edges  are  to  each  other  as 
1:2:3.    Find  the  edges. 

10.  The  dimensions  of  a  rectangular  bin  are  4  ft.,  4^  ft.,  and 
10  ft.,  and  it  is  desired  to  treble  its  capacity.  How  can  this  be 
done  if  only  one  dimension  is  changed  ?  two  dimensions  ?  all 
three  dimensions  ? 

11.  Make  a  geometrical  application  of  the  equation  (x  +  yy  = 
x^  +  Sx^'^j  +  Sxif  +  if. 

12.  How  much  will  it  cost,  at  40  cents  per  cubic  yard,  to  dig 
an  open  ditch  80  rd.  long,  6  ft.  wide  at  the  top,  2^  ft.  wide  at 
the  bottom,  and  3  ft.  deep  ? 

13.  How  many  square  feet  of  lead  will  be  required  to  line 
a  rectangular  cistern  10  ft.  long,  7  ft.  wide,  and  4^  ft.  deep  ? 
What  will  be  the  weight  of  the  lead  if  it  is  ^^  in.  thick  and  a 
cubic  inch  weighs  .411  lb.  ? 

14.  What  is  the  weight  of  the  water  received  upon  an  acre 
of  ground  during  a  storm  in  which  rain  falls  to  the  depth  of 
an  inch  ? 

15.  Allowing  30  cu.  ft.  of  air  per  minute  for  each  person  in 
this  classroom,  how  much  air  must  be  driven  into  the  room 
and  how  many  times  must  the  air  be  changed  during  the 
recitation  period  to  insure  good  ventilation  ? 

16.  The  cross  section  of  a  trough  12  ft.  long  is  an  equilateral 
triangle.  When  20  gal.  of  water  are  poured  into  the  trough, 
whose  edges  are  in  the  same  horizontal  plane,  how  deep  will 
the  water  be  ? 

17.  A  room  is  10  ft.  high  and  its  length  is  one  half  greater 
than  its  width.  If  the  area  of  the  ceiling  and  walls  is  816  sq.  ft., 
find  the  other  two  dimensions. 


EXERCISES  IN  SOLID  GEOMETRY  179 

18.  A  block  of  ice  1^  ft.  by  2  ft.  by  3  ft.  is  placed  in  a  box 

4  ft.  long  and  2  ft.  wide.  What  will  be  the  depth  of  water  in 
the  box  after  the  ice  melts,  the  specific  gravity  of  ice  being 
.917  ? 

19.  How  large  a  cubical  reservoir  will  be  required  to  hold 
the  water  that  falls  on  the  roof  of  a  house  covering  548  sq.  ft. 
of  ground,  during  a  shower  in  which  |  of  an  inch  of  rain  falls  ? 

20.  How  many  square  yards  of  canvas  will  be  required  to 
make  a  tent  10  ft.  by  16  ft.,  if  the  sides  are  6  ft.  high  and  the 
roof  has  ^  pitch  ? 

21.  An  oblique  prism  whose  altitude  is  h  has  for  its  base  a 
rhombus  whose  diagonals  are  k  and  I.   Find  its  volume. 

22.  Two  rectangular  parallelepipeds  are  to  each  other  as 

5  :  18.  The  dimensions  of  the  first  are  5,  13^,  and  18.  Find 
the  dimensions  of  the  other,  if  they  are  to  each  other  as 
1:2:3. 

23.  The  base  of  a  prism  whose  altitude  is  15  cm.  is  a  quadri- 
lateral whose  sides  are  10  cm.,  18  cm.,  12  cm.,  and  16  cm.,  the 
last  two  forming  a  right  angle.    Find  its  volume. 

24.  A  prism  has  for  its  base  a  triangle  whose  sides  are  to 
each  other  as  5  :  12  :  13.  If  its  altitude  is  4  m.  and  its  volume 
is  4.8  cu.  m.,  find  the  sides  of  the  base. 

25.  The  Great  Pyramid  is  762  ft.  square  at  the  base  and  484  ft. 
high.   Compute  its  volume  and  its  lateral  area. 

26.  The  lateral  area  of  a  regular  square  pyramid  of  wood  is 
144  sq.  in.,  and  one  side  of  the  base  is  8  in.  Find  its  weight, 
if  its  specific  gravity  is  .53. 

27.  Determine  the  volume  of  a  pyramid,  one  of  whose  lateral 
faces  is  an  equilateral  triangle  on  a  side  of  18  in.,  and  whose 
third  lateral  edge  is  perpendicular  to  the  other  two  and  is 
24  in.  long. 

28.  A  section  of  a  pyramid  parallel  to  the  base  contains 
96  sq.  ft.,  and  its  distance  from  the  base  whose  area  is  120  sq.  ft. 
is  4  ft.    Find  the  altitude  of  the  pyramid. 


180  APPLIED  MATHEMATICS 

29.  The  lateral  area  of  a  regular  triangular  pyramid  is 
64  sq.  ft.  and  one  side  of  the  base  is  8  ft.    Find  the  altitude. 

30.  If  a  section  of  a  pyramid  parallel  to  the  base  is  so  taken 
that  its  area  is  ^  that  of  the  base,  what  part  of  the  pyramid  is 
that  portion  above  the  section  ? 

31.  If  the  sides  of  the  base  of  a  pyramid  are  4,  6,  7,  and  9, 
and  the  solid  is  cut  by  a  plane  parallel  to  the  base  so  that  the 
section  is  ^^  of  the  base,  what  will  be  the  lengths  of  the  sides 
of  the  section  ? 

32.  A  granite  obelisk  in  the  form  of  a  frustum  of  a  regular 
quadrangular  pyramid,  surmounted  by  a  pyramid  of  slant 
height  15  in.,  has  each  side  of  one  base  1  ft.  4  in.  and  each  side 
of  the  other  base  2  ft.  3  in.,  and  the  slant  height  is  12  ft.  If 
the  specific  gravity  of  the  granite  is  2.6,  find  the  weight  of  the 
obelisk. 

33.  What  will  be  the  expense  of  polishing  the  faces  of  the 
obelisk  in  the  preceding  exercise  at  50  cents  per  square  foot  ? 

34.  What  is  the  capacity  in  gallons  of  a  reservoir  12  ft.  in 
depth  and  300  ft.  long  by  160  ft.  wide  at  the  top,  the  slope  of 
the  walls  being  3:2? 

35.  A  granite  monument  in  the  form  of  a  prismoid  is  16  ft. 
high  and  the  dimensions  of  its  ends  are  42  in.  by  28  in.  and 
18  in.  by  12  in.  respectively.  What  is  its  weight  if  the  specific 
gravity  of  the  granite  is  2.7  ?  See  Kent's  ^'  Mechanical  Engi- 
neers' Pocket-Book  "  for  the  definition  of  a  prismoid  and  for 
the  prismoid  formula. 

36.  A  milldam  of  earth  with  plane  sloping  sides  and  rec- 
tangular bases  is  80  m.  by  6  m.  at  the  top  and  66  m.  by  18  m. 
at  the  bottom.    If  its  height  is  5.4  m.,  find  its  cubic  contents. 

37.  How  many  cubic  yards  of  earth  will  it  be  necessary  to 
remove  in  making  a  cut  for  a  railroad,  which  must  be  14  ft.  deep, 
24  ft.  wide,  240  ft.  long  at  the  bottom,  and  170  ft.  long  at  the 
top,  the  slope  of  the  sides  being  7  :  10  ? 


EXERCISES  IN  SOLID  GEOMETRY  181 

38.  Apply  the  prismoid  formula  to  the  regular  octahedron 
whose  edge  is  e. 

39.  The  volume  of  a  wedge  whose  base  is  7.5  cm.  by  12  cm., 
and  whose  height  is  3.5  cm.,  is  142  ccm.  Find  the  length  of  its 
edge,  regarding  the  solid  as  a  prismoid. 

40.  Find  the  weight  of  a  steel  wedge  whose  base  measures 
3  in.  by  7  in.,  the  edge  5  in.,  and  the  height  6  in.,  if  a  cubic 
inch  of  steel  weighs  .283  lb. 

41.  If  a  cubic  foot  of  steel  weighs  490  lb.,  what  is  the  weight 
of  a  hollow  steel  beam  10  in.  square  at  one  end,  7  in.  at  the 
other  end,  and  18  ft.  long,  the  metal  being  |  in.  thick  ? 

42.  Find  the  cost  of  painting  the  lateral  surface  of  an  octag- 
onal tower  whose  slant  height  is  40  ft.,  if  the  short  diameter 
of  the  lower  base  is  12  ft.  and  of  the  upper  base  3  ft.,  at  24 
cents  per  square  yard. 

43.  At  what  distance  from  the  vertex  of  any  pyramid  must 
a  lateral  edge  12  ft.  long  be  cut  by  planes  parallel  to  the  base, 
in  order  that  the  areas  of  the  sections  formed  may  be  to  each 
other  as  2  :  3  :  5  ? 

44.  What  must  be  the  height  of  a  prism  of  iron  equal  in 
weight  to  the  sum  of  three  other  prisms  of  iron  of  the  same 
shape,  the  height  of  the  latter  being  2  in.,  3  in.,  and  4  in.  re- 
spectively ? 

45.  A  block  of  granite  weighs  2  T.  and  its  width  is  3  ft. 
What  is  the  width  of  a  block  of  granite  of  the  same  shape 
whose  weight  is  6  T.  ? 

46.  A  block  of  wood  of  specific  gravity  .675  weighs  72.4  lb., 
and  a  block  of  steel  of  specific  gravity  7.84  and  of  the  same 
shape  weighs  13.14  lb.  Find  the  ratio  of  their  corresponding 
dimensions. 

47.  Of  two  bodies  of  the  same  form,  one  weighs  2  lb.  and 
its  specific  gravity  is  .24,  while  the  other  weighs  56  lb.  and  itS 
specific  gravity  is  2.32.  If  one  dimension  of  the  first  body  is 
60  cm.,  what  is  the  corresponding  dimension  of  the  second  ? 


182  APPLIED  MATHEMATICS 

48.  An  irregular  mass  of  iron,  specific  gravity  7.2,  weighs 
42^  lb.  What  is  the  weight  of  a  mass  of  gold  of  the  same  form, 
specific  gravity  19.3,  if  two  corresponding  lines  of  the  two 
masses  have  the  ratio  2:3? 

49.  How  much  tin  will  be  required  to  make  an  open  cylin- 
drical vessel  of  altitude  65  cm.  which  shall  contain  160.2 1., 
taking  no  account  of  seams  ? 

50.  What  is  the  amount  in  cubic  feet  of  evaporation  daily 
from  a  circular  fishpond  6  rd.  in  diameter,  if  the  loss  in  depth 
is  .04  in.  ? 

51.  How  many  board  feet  of  lumber  16  in.  wide  can  be  made 
from  a  round  log  20  in.  in  diameter  and  16  ft.  long  ? 

52.  The  areas  of  two  sections  of  a  cylinder  of  revolution 
4  ft.  high,  which  are  parallel  to  the  axis  and  to  each  other,  are 
6  sq.  ft.  and  4§  sq.  ft.  respectively.  If  the  sections  are  2  in. 
apart,  what  is  the  volume  of  the  cylinder  ? 

53.  If  a  cubic  foot  of  copper  is  drawn  into  a  wire  ^^  in.  in 
diameter,  what  will  be  its  length  ? 

54.  An  irregTilar  mass  weighing  21.07  kg.  is  dropped  into  a 
cylindrical  vessel  42  cm.  in  diameter,  partially  filled  with  water. 
If  the  water  rises  80  cm.,  find  the  volume  and  specific  gravity 
of  the  body. 

55.  How  many  cubic  yards  of  stone  will  be  required  for  a 
semicircular  culvert  under  a  railroad  bank  112  ft.  wide,  the 
throat  of  the  culvert  being  6  ft.  high  and  the  walls  2  ft.  thick  ? 

56.  A  hollow  cylindrical  iron  column  is  14  ft.  4  in.  long,  6  in. 
in  diameter,  and  1  in.  thick.  What  is  its  weight  if  the  specific 
gravity  of  iron  is  7.2  ? 

57.  A  steel  shaft  is  reduced  in  diameter  in  a  lathe  from 
6  in.  to  4.5  in.  Find  to  the  nearest  hundredth  what  part  of  its 
weight  is  lost. 

58.  In  what  time  will  a  1-in.  circular  pipe  in  which  a  flow 
of  water  of  1  ft.  per  second  is  maintained,  fill  a  rectangular 
cistern  of  dimensions  3^  ft.  by  4  ft.  by  7^  ft.  ? 


EXERCISES  IN  SOLID  GEOMETRY  183 

59.  The  plunger  of  a  certain  single-acting  pump  is  10  in.  in 
diameter,  has  a  10-in.  stroke,  and  makes  15  strokes  per  minute. 
How  many  gallons  of  water  pass  through  it  in  12  hr.  ? 

60.  A  HoUey  pump  has  an  hourly  capacity  of  145,800  gal. 
of  water.  If  the  plunger  has  a  40-in.  stroke  and  makes  18  strokes 
per  minute,  what  is  its  diameter  ? 

61.  When  a  pump  is  required  to  furnish  2,800,000  gal.  of 
water  in  24  hr.,  how  many  strokes  per  minute  must  the  plunger 
make  if  its  diameter  is  30  in.  and  its  stroke  is  40  in.  ? 

The  following  rule  is  sometimes  used  to  calculate  the  horse  power 
of  a  steam,  boiler.  To  the  heating  surface  afforded  by  the  flues  is  to 
be  added  two  thirds  of  the  lateral  surface  of  the  boiler,  and  two 
thirds  of  one  flue  sheet  diminished  by  the  ends  of  the  flues.  In  gen- 
eral practice  12  sq.  ft.  of  heating  surface  are  considered  to  afford 
1  h.  p. 

62.  Compute  the  horse  power  of  a  steam  boiler  whose  length 
is  16  ft.  and  diameter  6  ft.,  if  there  are  136  flues,  each  16  ft. 
long  and  3  in.  in  interior  diameter. 

63.  What  must  be  the  length  of  the  flues  of  a  steam  boiler 
of  diameter  2  ft.,  containing  34  2-in.  flues,  in  order  that  it  may 
afford  12  h.  p.  ? 

64.  A  conical  heap  of  grain  4  ft.  high  covers  a  space  12  ft.  in 
diameter  on  the  floor.  How  large  must  be  a  cubical  bin  to 
hold  it  ? 

65.  How  many  square  yards  of  canvas  are  required  to  make 
a  conical  tent  10  ft.  high,  such  that  a  man  6  ft.  tall  may  stand 
without  stooping  anywhere  within  4  ft.  of  the  center  ? 

66.  A  conical  vessel  whose  angle  is  60°  is  filled  to  the  depth 
of  8  in.  with  water,  and  when  a  solid  cube  of  wood  is  submerged 
in  it,  the  water  rises  1  in.    Find  the  edge  of  the  cube. 

67.  How  much  ground  is  covered  by  a  conical  tent  9  ft.  in 
height,  which  contains  162  sq.  ft.  of  canvas  ? 

68.  A  square  whose  side  is  4  cm.  revolves  around  one  of  its 
diagonals.   Find  the  volume  generated. 


184  APPLIED  MATHEMATICS 

69.  A  rectangle  6  in.  by  8  in.  revolves  around  one  of  its 
diagonals.  Determine  the  volume  and  the  area  of  the  surface 
generated. 

70.  If  a  sector  of  120°  is  cut  out  of  a  circular  piece  of  canvas 
28  ft.  in  diameter,  what  are  the  dimensions  of  the  conical  tent 
that  can  be  made  out  of  the  remainder  ? 

71.  A  hollow  iron  cone  is  4  in.  long  and  4  in.  in  diameter, 
and  the  metal  is  ^  in.  thick.  Find  its  weight  if  a  cubic  inch  of 
iron  weighs  .261  lb. 

72.  The  altitudes  of  a  cylinder  and  an  equivalent  cone  are  to 
each  other  as  16  :  27.    Find  the  ratio  of  their  other  dimensions. 

73.  At  15  cents  per  square  foot,  what  will  be  the  cost  of 
cementing  the  walls  and  bottom  of  a  cistern  in  the  form  of  an 
inverted  frustum  of  a  cone  of  revolution  whose  depth  is  7  ft. 
and  diameters  6  ft.  and  3  ft.,  the  lid  1^  ft.  square  not  being 
cemented  ? 

74.  A  cone  whose  slant  height  is  16  cm.  is  to  be  divided  into 
three  parts  in  the  ratio  of  1 :  2  :  3.  At  what  distance,  measuring 
from  the  vertex,  must  the  slant  height  be  cut  by  planes  parallel 
to  the  base  ? 

75.  In  a  sphere  of  radius  5  ft.,  what  is  the  area  of  the  circle 
whose  plane  is  4  ft.  from  the  center  ? 

76.  In  a  sphere  of  radius  6  ft.  how  far  from  the  center  is 
the  plane  of  a  circle  whose  area  is  50f  sq.  ft.  ? 

77.  What  is  the  length  of  an  arc  of  120°  in  the  circumference 
of  a  circle  whose  plane  is  4^  ft.  from  the  center  of  a  sphere  of 
radius  5  f t.  ? 

78.  On  a  sphere  &f  radius  6  in.  what  is  the  polar  distance  of 
a  small  circle  whose  latitude  is  60°  ?  What  is  the  radius  of  the 
circle  ? 

79.  How  many  degrees  in  each  angle  of  an  equilateral 
spherical  triangle  whose  area  is  j^^  of  that  of  the  sphere  ? 

80.  If  a  birectangular  triangle  is  ^^  of  the  surface  of  its 
sphere,  what  is  the  third  angle  of  the  triangle  ? 


EXERCISES  IN  SOLID  GEOMETRY  185 

81.  If  the  diameter  of  the  moon  is  2162  mi.,  find  its  surface 
in  square  miles  and  its  volume  in  cubic  miles. 

82.  The  diameter  of  the  earth  is  7918  mi.  and  that  of  the 
planet  Mercury  3030  mi.  If  the  density  of  the  latter  is  2.23 
times  that  of  the  former,  show  that  the  mass  of  Mercury  is 
nearly  ^  that  of  the  earth. 

83.  If  the  mean  diameters  of  the  earth  and  the  moon  are 
7918  and  2162  mi.  respectively,  show  that  the  ratio  of  their 
surfaces  is  27  :  2  nearly. 

84.  What  is  the  diameter  of  a  sphere  of  which  a  wedge  of 
11°  15'  contains  359.3  cu.  dm.  ? 

85.  How  many  bullets  1  in.  in  diameter  can  be  made  of  3  ft. 
of  lead  pipe  1^  in.  in  exterior  diameter  and  ^  in.  thick  ? 

86.  A  steel  ball  6  in.  in  diameter  is  dropped  into  a  cylindrical 
vessel  8  in.  in  diameter,  which  is  filled  within  2  in.  of  the  top 
with  water.    How  much  water  will  overflow  ? 

87.  If  400  lead  balls  each  ^  in.  in  diameter  are  melted 
and  run  into  a  disk  ^^  in.  thick,  what  will  be  the  radius  of 
the  disk? 

88.  How  many  bullets  of  caliber  .32  (.32  in.  in  diameter)  can 
be  made  from  a  bar  of  lead  2i  in.  by  4  in.  by  6  in.  ? 

89.  A  marble  |  in.  in  diameter  is  dropped  into  a  conical  glass 
whose  diameter  is  2  in.  and  depth  3  in.,  and  is  just  covered  by 
the  water  that  it  contains.  What  was  the  depth  of  the  water 
at  first  ? 

90.  What  is  the  altitude  of  that  zone  of  a  sphere  which 
equals  a  trirectangular  triangle  in  area  ? 

91.  Find  the  surface  of  the  zone  of  a  sphere  of  radius  8  in. 
cut  off  by  a  plane  3  in.  from  the  center  of  the  sphere. 

92.  What  is  the  altitude  of  a  zone  of  120  sq.  in.  surface,  if 
the  radius  of  the  sphere  is  10  in.  ? 

93.  How  far  from  the  surface  of  a  sphere  must  a  lamp  be 
placed  in  order  that  one  sixth  of  the  surface  may  be  illmninated? 


186  APPLIED  MATHEMATICS 

94.  Show  that  the  portion  of  the  earth's  surface  that  is 

visible  to  an  aeronaut  at  a  height  h  above  the  surface  is —  > 

^  r  +  h 

r  being  the  radius  of  the  earth.  When  h  is  small  it  may 
be  dropped  in  the  denominator,  giving  the  approximate  area 
2  7rrA. 

95.  On  a  globe  of  radius  7  cm.  it  is  desired  to  mark  off  a 
zone  whose  area  shall  be  6.16  sq.  cm.  What  opening  of  the 
compasses  shall  be  used  ? 

96.  On  a  globe  of  radius  9  in.  a  small  circle  is  described 
with  an  opening  of  the  compasses  of  6  in.  Find  the  length  of 
the  circumference. 

97.  The  altitude  and  radius  of  the  base  of  a  right  cone 
are  12  and  9  in.  respectively.  Find  the  radius  of  the  circle  of 
tangency  of  the  inscribed  sphere. 

98.  How  does  the  specific  gravity  of  a  spherical  body  com- 
pare with  that  of  a  liquid  in  which  it  floats,  with  one  half  its 
surface  above  the  surface  of  the  liquid  ?  one  third  ?  when  it  is 
just  submerged  ? 

99.  If  a  sphere  of  oak  6  in.  in  diameter  floats  in  water  with 
.3  of  its  surface  above  the  surface  of  the  water,  what  is  the 
specific  gravity  of  the  oak  ? 

100.  What  portion  of  the  surface  of  a  ball  of  iron  of  diameter 
1  in.  and  specific  gravity  7.2  will  remain  visible  when  it  is 
dropped  into  a  dish  of  mercury  whose  specific  gravity  is  13.6  ? 

II.    Graphical  Exercises 

99.  A  few  of  these  exercises  should  be  worked  out  carefully 
in  the  notebook. 

1.  Construct  a  graph  to  show  the  change  in  the  volume  of  a 
cube  as  its  edge  increases  from  0  to  12  in.  What  is  the  equation 
of  the  graph  ? 


EXERCISES  IN  SOLID  GEOMETRY 


187 


2.  On  the  same  axes  as  in  Exercise  1  show  graphically  the 
change  in  the  surface  of  the  cube.  How  do  the  graphs  show 
(a)  when  the  surface  equals  the  volume  numerically  ?  (b)  when 
a  cube  has  a  greater  surface  than  volume  ?  Write  on  each  graph 
its  equation. 


Y 

i355- 

/ 

IZOO 

h 

''JP 

1100 

Au 

TITU 

DE 

0 

2 

A 

6 

e 

M 

IZ 

1 

1000           1 
Volume: 

0 

6 

AS 

I6Z 

384 

750 

itee 

/ 

90O  {         1 
Surface. 

0 

15 

60 

135 

tAO 

375 

540 

/ 

BOO 

j\ 

f 

700 

/ 

600 

J 

/ 

y 

Surface 

5oo 

/ 

y 

y 

400 

A 

A 

V 

kx) 

/ 

A 

^ 

eo6 

> 

I' 

— 

^ 

:y' 

lOO 

_^ 

^ 

iH 

— 

X 

5*^ 

^ 







— 

- — 





'      "■ 

^ 

~__ 

> 

0 

' 

^ 

3 

A 

5^' 

rituc 

6 

e 

|7 

6 

9 

lO 

,, 

IZ. 

Fig.  80 


3.  The  altitude  of  a  regular  square  pyramid  is  12  ft.  and 
each  side  of  the  base  is  18  ft.  Show  graphically  (a)  the  volume, 
(i)  the  lateral  surface  of  the  pyramids  cut  off  from  the  vertex 
by  planes  parallel  to  the  base.  Find  the  ratio  of  the  surface 
of  any  of  the  pyramids  to  its  voluine,  and  use  the  result  to  check 
the  table  of  values. 


188  APPLIED  MATHEMATICS 

4.  The  altitude  of  a  right  cone  is  12  ft.  and  the  radius  of 
the  base  is  9  ft.  Show  graphically  (a)  the  volume,  (b)  the  lat- 
eral surface  of  the  cones  cut  off  from  the  vertex  by  planes 
parallel  to  the  base. 

5.  On  the  same  axes  construct  graphs  to  show  the  change 
(a)  in  volume,  (b)  in  lateral  surface  of  a  right  cylinder  the  radius 
of  whose  base  is  6  in.,  as  its  altitude  increases  from  0  to  15  in. 

6.  Represent  graphically  the  change  in  the  area  of  a  section 
parallel  to  the  base  of  a  regular  triangular  pyramid,  the  side 
of  whose  base  is  8  cm.  and  whose  altitude  is  12  cm. 

7.  On  the  same  axes  represent  graphically  the  change  (a)  in 
volume,  (b)  in  surface  of  a  sphere  as  the  radius  increases  from 
0  to  10  in. 

8.  The  volume  of  a  pyramid  is  60  cu.  in.  Construct  a  graph 
to  show  the  relation  between  the  base  and  altitude  as  the  altitude 
increases  from  0  to  180  in. 

9.  The  volume  of  a  cylinder  is  440  cu.  in.  Construct  a  curve 
to  show  the  relation  between  the  radius  of  the  base  and  the 
altitude,  as  the  radius  increases  from  0  to  10  in. 

10.  From  each  corner  of  a  square  piece  of  tin  12  in.  on  a 
side  a  smaller  square  is  cut,  the  remainder  of  the  sheet  being 
bent  so  as  to  form  a  rectangular  open  box.  Determine  the  side 
of  each  small  square  in  order  that  the  capacity  of  the  box  may 
be  as  great  as  possible. 

11.  If  the  sheet  of  tin  in  the  preceding  exercise  had  been 
rectangular,  20  in.  by  12  in.,  what  then  would  have  been  the 
size  of  each  small  square  ? 

12.  A  bin  with  a  square  base  and  open  at  the  top  is  to  be 
constructed  to  contain  400  cu.  ft.  of  grain.  What  must  be  its 
dimensions  to  require  the  least  amount  of  material  ? 

13.  A  closed  cylindrical  oil  tank  is  required  to  hold  100  bbl., 
each  of  42  gal.  What  dimensions  will  necessitate  the  least 
steel  plate  in  the  making  ? 


EXERCISES  IN  SOLID  GEOMETRY  189 

14.  An  open  rectangular  tank  whose  length  is  to  be  twice 
its  width  is  to  hold  200  gal.  of  water.  What  dimensions  will 
require  the  least  amount  of  lining  for  the  tank  ? 

15.  The  strength  of  a  rectangular  beam  is  proportional  to 
the  product  of  its  breadth  and  the  square  of  its  depth.  What 
are  the  dimensions  of  the  strongest  beam  that  can  be  cut  from 
a  round  log  2  ft.  in  diameter  ? 

16.  If  the  slant  height  of  a  right  cone  is  12  ft.,  what  must 
be  the  radius  of  its  base  in  order  that  its  volume  may  be  as 
great  as  possible  ? 

17.  Determine  the  right  cylinder  of  greatest  lateral  surface 
that  can  be  inscribed  in  a  cone  of  revolution  whose  altitude  is 
14  in.  and  radius  of  base  8  in. 

18.  Find  the  dimensions  of  the  smallest  cone  of  revolution 
that  can  be  circumscribed  about  a  cylinder  whose  altitude  and 
radius  are  respectively  9  dm.  and  3  dm. 

19.  The  stiffness  of  a  rectangular  beam  varies  as  the  product 
of  its  breadth  and  the  cube  of  its  depth.  Find  the  dimensions  of 
the  stiffest  beam  that  can  be  sawed  from  a  log  20  in.  in  diameter. 

20.  Determine  the  dimensions  of  the  largest  right  cone  that 
can  be  inscribed  in  a  sphere  of  radius  5  in. 

21.  Find  what  radius  of  the  base  of  a  conical  tent  of  375  cu.  ft. 
capacity  will  require  the  least  amount  of  canvas  in  the  making. 
Also  find  the  relation  between  the  altitude  and  the  radius. 

22.  Find  the  radius  of  the  right  cylinder  of  greatest  lateml 
surface  that  can  be  inscribed  in  a  sphere  whose  diameter  is  12  in. 

23.  Find  the  relation  between  the  radius  of  the  base  and 
the  altitude  of  a  right  cone  whose  convex  surface  contains 
264  sq.  ft.,  in  order  that  the  volume  may  be  as  great  as  possible. 

24.  Determine  the  altitude  of  the  least  cone  of  revolution 
that  can  be  circumscribed  about  a  sphere  of  radius  2  dm. 

25.  What  must  be  the  altitude  of  the  cone  of  revolution  of 
least  lateral  surface  that  can  be  circumscribed  about  a  sphere 
whose  radius  is  4  in.  ? 


190  APPLIED  MATHEMATICS 

III.   Algebraic  Problems 

100.  Make  a  sketch  for  each  problem.  Put  the  given  dimen- 
sions on  the  figure  and  set  up  the  equations  from  the  sketch. 

1.  What  are  the  other  two  dimensions  of  a  rectangular 
parallelepiped  whose  length  is  8  in.,  if  its  volume  is  160  cu.  in. 
and  its  total  surface  is  184  sq.  in.  ? 

2.  If  the  three  face  diagonals  of  a  rectangular  solid  are 
respectively  6,  7,  and  9  cm.,  what  must  be  the  dimensions  of 
the  solid  ? 

3.  One  dimension  of  a  rectangular  parallelepiped  is  6  in., 
one  diagonal  is  12  in.,  and  the  area  of  one  of  the  wholly  unknown 
faces  is  44  sq.  in.    What  are  the  other  two  dimensions  ? 

4.  The  sum  of  the  three  dimensions  of  a  rectangular  solid 
is  12  and  the  diagonal  of  the  solid  is  5  V2.  Find  its  total  surface. 

5.  The  sum  of  a  diagonal  and  an  edge  of  a  cube  is  6.  Find 
an  edge  of  the  cube. 

6.  The  area  of  one  face  of  a  rectangular  solid  is  10  sq.  cm., 
that  of  another  is  15  sq.  cm.,  and  the  total  area  is  100  sq.  cm. 
Find  the  dimensions. 

7.  What  are  the  dimensions  of  a  rectangular  solid  whose 
entire  surface  is  392  sq.  in.,  if  its  top  contains  96  sq.  in.  and 
one  end  40  sq.  in.  ? 

8.  Given  the  diagonal  of  a  cube  equal  to  k.  Find  the  volume 
of  the  cube  and  its  surface. 

9.  Given  the  volume  v  and  the  altitude  A  of  a  regular  hexag- 
onal prism.    Find  s,  the  length  of  One  side  of  the  base. 

10.  The  sides  of  the  base  of  a  triangular  prism  are  as  3  :  4:5, 
and  its  volume  is  432  cu.  ft.  If  the  altitude  is  4  ft.,  find  the 
sides  of  the  base. 

11.  What  must  be  the  altitude  of  a  pyramid  in  order  that 
its  total  area  may  be  equal  to  the  sum  of  the  areas  of  two  similar 
pyramids  whose  altitudes  are  respectively  6  and  4  in.  ? 


EXERCISES  IN  SOLID  GEOMETRY  191 

12.  What  is  the  altitude  of  a  pyramid  whose  base  contains 
98  sq.  in.,  if  a  section  parallel  to  the  base  and  4  in.  from  the 
vertex  contains  32  sq.  in.  ? 

13.  The  volume  of  a  pyramid  with  a  rectangular  base  is  76.8 
cu.  in.,  one  side  of  the  base  is  9.6  in.,  and  the  altitude  exceeds 
the  other  side  of  the  base  by  2  in.  Find  the  altitude  and  the 
other  side  of  the  base. 

14.  If  a  square  pyramid  has  each  basal  edge  equal  to  e  and 
each  lateral  edge  equal  to  e^,  show  that  the  volume  will  be 

y  =  I  V2(2ef-e2). 

15.  Given  v,  the  volume,  and  s,  one  side  of  the  square  base 
of  a  regular  quadrangular  pyramid,  find  the  lateral  surface. 

16.  Derive  an  expression  for  the  volume  of  a  regular  tetra- 
hedron in  terms  of  its  edge  e. 

17.  An  iron  plate  8  in.  long  and  1\  in.  thick  has  squared  ends 
but  uniformly  and  equally  beveled  sides,  and  contains  122  cu.  in. 
If  the  difference  of  the  widths  of  the  two  flat  faces  is  2.8  in., 
find  those  widths. 

18.  The  lateral  area  of  a  frustum  of  a  regular  quadrangular 
pyramid  is  281.2  sq.  in.,  the  slant  height  is  15.2  in.,  and  a  side 
of  the  lower  base  exceeds  a  side  of  the  upper  base  by  3.75  in. 
Find  a  side  of  each  base. 

19.  What  must  be  the  diameter  of  a  cylindrical  gas  holder 
which  is  to  hold  6,000,000  feet  of  gas,  if  its  height  is  to  be  f 
of  its  diameter  ? 

20.  The  sum  of  the  numerical  measures  of  the  volume  and 
lateral  area  of  a  cylinder  of  revolution  is  231.  If  the  altitude 
is  14,  what  is  the  diameter  ? 

21.  Write  the  formula  that  gives  t,  the  total  surface  of  a 
cylinder  of  revolution,  in  terms  of  A,  the  altitude,  and  r,  the 
radius  of  the  base,  and  solve  it  for  h  and  r. 

In  case  of  a  cylinder  of  revolution : 

22.  Given  t  and  r,  find  A  and  v. 


192  APPLIED  MATHEMATICS 

23.  Given  v  and  r,  find  h  and  t. 

24.  Given  v  and  A,  find  r  and  I  (the  lateral  area), 

25.  Given  I  and  v,  find  h,  r,  and  ^. 

26.  Given  I  and  7i,  find  r,  v,  and  ^. 

27.  Given  t  and  v,  find  7;  h,  and  Z. 

Suggestion.  Find  r  by  trial  from  2  7rr'  —  /?•  +  2  i'  =  0  for  any  given 
numerical,  values  of  t  and  v  (see  sect.  58)  ;  then  find  h  from  v  —  irr^h, 
and  then  /  from  1  =  2  irrh. 

28.  How  far  from  the  axis  of  a  cylinder  of  revolution  whose 
height  is  h  ft.  and  diameter  d  ft.  must  a  plane  parallel  to  the 
axis  be  passed,  in  order  to  make  a  section  of  area  k  sq.  ft.  ? 

29.  If  the  total  surface  of  a  cone  of  revolution  is  21  tt  and 
the  slant  height  is  4,  find  the  radius  and  the  volume  of  the  cone. 

30.  The  sum  of  the  altitude  and  the  radius  of  the  base  of  a 
cone  of  revolution  is  11  and  their  product  is  10.  What  is  the 
volume  of  the  cone  ? 

31.  The  lateral  area  of  a  right  cone  is  9  ViOtt,  and  its  alti- 
tude is  equal  to  3  times  the  radius  of  its  base.    Find  its  volume. 

32.  Find  the  slant  height  and  the  radius  of  the  base  of  a 
cone  of  revolution  whose  total  surface  is  462  sq.  in.,  and  the 
sum  of  the  slant  height  and  the  radius  is  21  in. 

33.  The  lateral  surface  of  a  right  cone  whose  slant  height  is 
6  exceeds  the  base  by  12^.    Find  the  radius  of  the  base. 

34.  What  is  the  radius  of  the  upper  base  of  a  frustum  of  a 
right  cone,  if  its  volume  is  .516  it  cu.  dm.,  its  altitude  1.2  dm., 
and  the  radius  of  its  lower  base  .8  dm.  ? 

35.  The  lateral  area  of  a  frustum  of  a  cone  of  revolution  is 
77  TT,  the  slant  height  is  7,  and  the  altitude  is  2  Vo.  Find  the 
radii  of  the  bases. 

36.  What  is  the  volume  of  a  frustum  of  a  right  cone  the  sum 
of  the  radii  of  whose  bases  is  11  and  their  product  28,  the 
altitude  being  7  ? 


EXERCISES  IN  SOLID  GEOMETRY  193 

37.  Find  the  radii  of  the  bases  of  a  frustum  of  a  right  cone, 
given  the  lateral  area  as  1068^  sq.  ft.,  the  slant  height  as  17  ft., 
and  the  altitude  as  15  ft. 

38.  The  volumes  of  two  spheres  are  to  each  other  as  8  :  125, 
and  the  sum  of  their  radii  is  12  in.    Find  the  radii. 

39.  The  product  of  the  radii  of  two  spheres  is  22.5  and  the 
ratio  of  their  surfaces  is  25  :  64.    What  are  the  radii  ? 

40.  If  the  surface  of  a  sphere  is  equal  to  the  sum  of  the 
surfaces  of  two  spheres  whose  radii  are  2  in.  and  4  in.  respec- 
tively, how  does  its  volimie  compare  with  the  sum  of  their 
volumes  ? 

41.  What  is  the  radius  of  a  sphere  of  which  a  zone  of 
24  sq.  in.  is  illuminated  by  a  lamp  placed  18  in.  from  its 
surface  ? 

42.  What  relation  must  the  radius  of  a  given  sphere  bear  to 
the  radii  of  two  other  spheres  if  its  surface  is  a  mean  propor- 
tional between  their  surfaces  ? 

43.  Compare  the  expression  for  the  volume  of  a  sphere  with 
that  for  its  surface,  and  determine  how  long  the  radius  must 
be  in  order  that  the  volume  may  be  nimierically  greater  than 
the  surface. 

44.  In  a  sphere  of  radius  8  the  radius  of  one  small  circle  is 
a  mean  proportional  between  the  radius  of  the  sphere  and  the 
radius  of  another  small  circle,  and  the  sum  of  the  radii  of  the 
two  small  circles  is  10.    Find  the  radii  of  the  small  circles. 

45.  Derive  an  expression  in  one  variable  for  the  volume  of 
a  right  cone  inscribed  in  a  sphere  of  radius  r. 

46.  Find  an  expression  in  terms  of  the  altitude  for  the  total 
surface  of  a  cylinder  of  revolution  inscribed  in  a  sphere  of 
radius  r. 

47.  What  is  the  expression  for  the  volume  of  a  right  cylinder 
inscribed  in  a  right  cone,  altitude  h,  radius  of  base  r,  in  terms 
of  the  radius  of  the  cylinder  ? 


194  APPLIED  MATHEMATICS 

48.  Find  an  expression  in  one  variable  for  the  total  surface 
of  a  right  cone  circumscribed  about  a  given  right  cylinder. 

49.  What  expression  in  one  variable  denotes  the  volume  of 
a  right  cone  circumscribed  about  a  given  sphere  ? 

50.  Derive  the  expression  in  one  variable  for  the  lateral 
surface  of  the  cone  in  Exercise  49. 

51.  Find  an  expression  in  one  variable  for  the  volume  of  a 
right  cone  circumscribed  about  a  given  right  cylinder. 

Problems  45-51  furnish  good  exercises  in  maxima  and  minima 
by  giving  numerical  values  to  the  dimensions  of  the  constant  solids. 
Since  some  of  the  expressions  are  rather  complicated  the  work  of 
computing  the  table  of  values  may  be  divided  among  the  members 
of  the  class,  each  one  computing  the  value  of  the  function  for  a 
single  value  of  the  variable. 


CHAPTER  XVI 

HEAT 

101.  Thermometers.  Though  the  Fahrenheit  scale  is  in 
general  use  in  everyday  life  and  in  ordinary  engineering  work, 
the  Centigrade  scale  is  used  in  laboratories  and  all  scientific 
work  to  such  an  extent  that  one  should  become  acquainted  with 
it.  Fahrenheit  (Danzig,  Germany)  devised  his  scale  about 
1726.  He  thought  that  the  lowest  possible  degree  of  cold  was 
obtained  by  mixing  salt  and  ice ;  hence  he  took  as  zero  the 
position  of  the  mercury  when  placed  in  such  a  mixture.  It  is 
not  known  why  he  marked  the  boiling  point  of  water  212°. 
The  Centigrade  scale  was  proposed  by  Anders  Celsius  (Upsala, 
Sweden)  about  1741. 

In  the  Fahrenheit  thermometer  the  boiling  point  of  water  at 
sea  level  is  taken  at  212°  and  the  freezing  point  of  water  at  32°. 
In  the  Centigrade  thermometer  the  boiling  point  is  taken  at 
100°  and  the  freezing  point  at  0°.  Hence  180°  on  the  Fahrenheit 
scale  equals  100°  on  the  Centigrade  scale. 

180°  F.  =  100°  C. 

l°F.=  f°C.  (1) 

1°  C.  =  1°  F.  (2) 

It  should  be  remembered  that  a  division  on  the  Centigrade 
scale  is  longer  than  a  division  on  the  Fahrenheit  scale.  Hence 
in  changing  from  degrees  Centigrade  to  degrees  Fahrenheit  we 
get  a  greater  number  of  degrees,  and  from  Fahrenheit  to  Centi- 
grade we  get  a  smaller  number  of  degrees. 

Equations  (1)  and  (2)  enable  us  to  change  readily  from  one 
scale  to  the  other. 

196 


196 


APPLIED  MATHEMATICS 


ICO-- 

os- 


PROBLEMS 

1.  Construct  a  graph  to  change  a  number  of  degrees  of  one 
scale  to  degrees  of  the  other  scale.  Why  is  it  necessary  to 
locate  only  two  points  and  draw  a  straight  line  through  them  ? 

2.  Change  (a)  90°  F.  to  C. ;  (b)  200°  F.  to  C. ;  (c)  40°  C.  to 
F. ;  (d)  80°  C.  to  F. ;  (e)  150°  F.  to  C. ;  (/)  112°  F.  to  C.  Check 
by  the  graph. 

3.  The  sum  of  a  number  of  degrees  F.  and  a  number  of 
degrees  C.  is  121.  When  the  degrees  F.  are  changed  to  degrees 
C.  and  added  to  the  number  of  degrees  C.  the  result  is  85.  Find 
the  number  of  degrees  F.  and  C. 

4.  The  sum  of  a  number  of  degrees  F.  and  a 
number  of  degrees  C.  is  53.  If  each  number  of 
degrees  is  changed  into  the  other  scale,  the 
sum  is  73.    Find  the  number  of  degrees  F.  and  C. 

102.  To  change  thermometer  readings  from 
one  scale  to  the  other.  In  the  above  problems 
we  were  dealing  with  degrees  not  with  ther- 
mometer readings.  When  we  change  thermom- 
eter readings  from  one  scale  to  the  other  we 
must  take  account  of  the  difference  in  position 
of  the  zeros  on  the  two  scales. 

Thus  find  the  C.  reading  when  the  F.  reading 
is  80°.  Looking  at  Fig.  81,  we  see  that  by  tak- 
ing 32°  from  80°  we  get  48°,  the  number  of 
degrees  the  F.  reading  is  above  0°C.    Then 

48°F.  =  48°x  |C.  =  26.7°C. 

Similarly,  to  find  the  F.  reading  when  the  C. 
reading  is  70°, 

70°  C.  =  70°  X  f  F.  =  126°  F 

But  126°  F.  takes  us  only  to  32°  F.  opposite  0°  C.  Hence  we 
add  32°  to  ^et  the  F.  reading,  158°,  corresponding  to  70°  C 


C» 


(^ 


Fig.  81 


HEAT  197 

To  change  from  F.  to  C.  readings,  subtract  32°  ami  mMltiply 
the  difference  hy  ^.  To  change  from  C.  to  F.  readings,  multiply 
by  I  and  add  82°  to  the  product. 

C.  =  I  (F.  -  32°). 
F.  =  ^C. +  32°. 

103.  To  determine  the  relation  of  the  corresponding  read- 
ings of  the  two  thermometers  by  experiment.  Take  several 
readings  of  the  two  thermometers  on  different  days,  or  obtain 
the  readings  by  putting  the  thermometers  into  water  at  dif- 
ferent temperatures. 

Readings  obtained: 

F.         32°         47°         70°         96°         118°         151° 
CO  8  21  36  48  66 

Locate  these  points  on  squared  paper.  Units  :  C,  horizontal, 

1  large  square  =  10° ;   F.,  vertical,  1  large  square  =  10°.    On 

stretching  a  thread  along  these  points  it  will  be  found  that 

they  lie  nearly  in  a  straight  line.    Draw  a  straight  line  among 

the  points  so  that  they  are  distributed  evenly  above  and  below 

it.    This  line  is  the  graph  of  the  equation  which  connects  the 

corresponding  readings.    To  find  the  equation  we  will  suppose 

that  it  is  of  the  form       ^^  ^       , 

F.  =  a  C.  +  6,  (1) 

where  a  and  b  are  unknown  numbers  which  must  be  determined. 
Taking  the  second  and  fifth  points  and  substituting  the  read- 
ings  in  (1),  we  have        ^^  ^    ^^^^^ 

118  =  48a  +  ^». 

Solving  these  equations,  we  get  a  =  1.77,  &  =  32.8. 

Substituting  these  values  in  (1),  we  get  F.  =  1.77°  C.  +  32.8°. 

The  readings  in  the  experiment  were  not  taken  with  sufficient 
exactness  to  give  a  close  result  (see  sect.  59). 

Exercise.  Take  several  corresponding  readings  on  the  two 
thermometers  and  find  the  relation  as  above. 


198 


APPLIED  MATHEMATICS 


-PROBLEMS 

1.  Construct  a  graph  to  change  the  readings  of  one  ther- 
mometer to  those  of  the  other.  Units :  horizontal,  1  large 
square  =  20°  F. ;  vertical,  1  large  square  =  10°  C.  Take  the 
lower  left-hand  corner  as  the  origin  and  mark  it  —  40°.  Show 
that  —  40°  is  the  same  reading  on  both  scales.  Locate  one 
other  point.    Why  is  the  graph  a  straight  line  ? 

2.  Change  the  reading  of  one  thermometer  to  that  of  the 
other,  and  .check  by  the  graph : 

{a)  78°F.  toC.  (e)  195°F.  toC. 

{b)  18°F.toC.  (/)  -20°F.toC. 

(c)  88°C.  toF.  {g)  -30°C.  toF. 

{d)  60°  C.  to  F.  (A)  0°  F.  to  C. 

3.  The  melting  point  of  the  following  metals  is  given  in 
degrees  F.     Change  to  the  Centigrade  scale : 

Tin      ...  442°  to    446°  Copper     .     .     .  1929°  to  1996° 

Lead    .     .     .  608°  to    618°  Cast  iron      .     .  1922°  to  2075° 

Silver  .     .     .  1733°  to  1873°  Steel  ....  2372°  to  2532° 

Gold    .     .     .  1913°  to  2282°  Platinum     .     .  3227° 

4.  The  following  record  of  temperature  was  taken  from 
The  Chicago  Daily  News. 


3  P.M 

.     .     .69 

3  A.M 

....     67 

4  P.M 

...     68 

4  A.M 

....     66 

5  P.M 

...     68 

5  A.M 

....     65 

6pm          .... 

...     67 

6  A.M 

....     64 

7  P.M 

...     66 

7  A.M 

....     65 

8pm 

...     67 

8  A.M 

....     66 

9  P.M 

...     67 

9  A.M 

....     67 

10  P.M 

...     68 

10  A.M 

....     67 

11  P.M 

...     68 

11  A.M 

....     68 

12  midnight      .     .     . 

...     68 

12  noon   .... 

....     70 

1am 

...     69 

1  P.M 

....     73 

2  A.M 

.          fi8 

HEAT 


199 


Change  the  readings  to  Centigrade  by  using  the  graph,  and 
on  the  same  sheet  of  squared  paper  plot  a  curve  for  each  of 
the  two  sets  of  readings.    Are  the  curves  parallel  ?    Why  ? 

5.  What  temperature  is  expressed  by  the  same  number  on 
the  F.  and  C.  scales  ? 

6.  A  Fahrenheit  and  a  Centigrade  thermometer  are  placed 
in  a  liquid  and  the  F.  reading  is  found  to  be  double  the  C. 
reading.   What  is  the  temperature  of  the  liquid  in  degrees  C.  ? 

Expansion  of  Solids  by  Heat 

104.  Linear  expansion.  When  a  solid  is  subjected  to 
changes  of  temperature  its  length  changes ;  in  general,  the 
length  increases  as  the  temperature  rises,  and  decreases  as  it 
falls.  For  ordinary  temperatures  the  amount  of  change  is 
nearly  the  same  for  each  degree  of  increase  or  decrease.  The 
following  table  gives  results  that  have  been  secured  by  experi- 
ment ;  they  are  only  approximate. 


Linear  Expansion  of  Solids  for  1  Degree,  between 
0°  and  212°  F. 


Aluminum 
Brass,  plate 
Copper    .     .     . 
Glass,  white    . 
Iron,  wrought 
Iron,  cast    .     . 


.00001234 
.00001052 
.00000887 
.00000478 
.00000648 
.00000556 


Lead 00001571 

Platinum 00000479 

Steel,  cast 00000636 

Steel,  tempered    .     .     .00000689 

Tin 00001163 

Zinc 00001407 


The  amount  of  expansion  is  seen  to  be  very  small.  Thus 
when  we  say  that  the  linear  expansion  of  wrought  iron  is 
.00000648  we  mean  that  the  length  of  a  wrought-iron  rod 
100  ft.  long  increases  648  millionths  of  a  foot  when  the  tem- 
perature of  the  rod  rises  1  degree.  However,  provision  must  be 
made  for  this  expansion  in  all  construction  work ;  for  example, 
a  little  space  is  left  between  the  ends  of  the  rails  in  laying 
railway  track,  hot- water  pipes  have  telescopic  joints,  and  so  on. 


200  APPLIED  MATHEMATICS 

PROBLEMS 

1.  Find  the  linear  expansion  of  copper,  wrought  iron,  and 
tin  for  1°  C. 

2.  A  brass  wire  is  200  ft.  long  at  0°.    Find  its  length  at  85°. 

Solution.  200  x  .00001052  x  85  =  .179  ft. 

200  +  .179  =  200.179  ft.  =  200  ft.  2.2  in. 

3.  A  steel  chain  is  66  ft.  long  at  77°.  What  will  be  its 
length  at  32°? 

4.  The  iron  girders  of  a  railway  bridge  are  100  ft.  long  at 
a  temperature  of  10°.  What  will  be  the  length  of  the  girders 
at  90°  ? 

5.  A  lead  pipe  is  80  ft.  long  at  —  10°.  How  long  will  it  be 
at  100°  ? 

6.  A  brass  rod  is  5  m.  long  at  0°  C.  What  is  its  length  at 
38°  C? 

7.  What  is  the  length  of  a  wrought-iron  rod  at  0°  C.  if  it 
is  1.56  m.  long  at  100°  C.  ? 

8.  What  is  the  length  of  a  copper  wire  which  increases  in 
length  1.2  in.  when  its  temperature  is  raised  200°  ? 

9.  What  is  the  area  of  a  brass  plate  at  100°  C.  which 
measures  8.35  cm.  by  4.16  cm.  at  0°C.  ? 

10.  One  brass  yardstick  is  correct  at  32°  and  another  at  68°. 
What  is  the  difference  in  their  lengths  at  the  same  temperature  ? 

11.  A  bar  of  metal  10  ft.  long  at  200°  increases  in  length  .31  in. 
when  heated  to  362°.    Find  the  expansion  of  1  ft.  for  1°. 

12.  A  plate-glass  window  is  10  ft.  by  12  ft.  How  much  will 
it  change  in  area  when  its  temperature  changes  from  —  20°  to 
90°,  if  its  linear  expansion  for  1°  is  .000005  ? 

13.  An  iron  steam  pipe  200  ft.  long  at  190°  ranges  in  tem- 
perature from  190°  to  —  4°.  What  must  be  the  range  of  motion 
of  an  expansion  joint  to  provide  for  the  change  in  length  ? 


HEAT 


201 


14.  A  platinum  wire  and  a  brass  wire  are  each  100  ft.  long 
at  30°.  How  much  must  they  be  heated  to  make  the  brass  wire 
1  in.  longer  than  the  platinum  wire  ? 

Suggestion.    Let       x  =  the  number  of  degrees. 

.00001052  X  100  X  12  a;  -  .00000479  x  100  x  12  x  x  =  1. 

15.  A  copper  bar  is  10  ft.  long.  What  must  be  the  length  of 
a  cast-iron  bar  in  order  that  the  two  may  expand  the  same 
amount  for  1°  ? 

16.  A  steel  tape  100  ft.  long  is  correct  at  32°.  On  a  day  when 
its  temperature  was  88°  a  line  was  measured  and  found  to  be 
1  mi.  long.  What  was  the  error  and  what  was  the  true  length 
of  the  line  ? 

17.  An  iron  casting  shrinks  about  ^  in.  per  linear  foot  when 
cooling  down  to  70°.    What  is  the  shrinkage  per  cubic  foot  ? 

18.  The  Chicago  and  Oak  Park  Elevated  Railway  is  about 
9  mi.  long  from  Wabash  Avenue  to  Oak  Park  Station.  What 
is  the  difference  in  the  length  of  the  rails  for  a  change  in 
temperature  from  -  20°  to  80°  ? 

19.  Construct  a  graph  to  show  the  expansion  of  a  steel  wire 
100  ft.  long  as  its  temperature  rises  from  0°  to  2000°. 

20.  The  following  table  shows  the  change  in  the  volmne  of 
water  as  its  temperature  rises  from  0°  to  17°  C.  Construct  the 
graph.  How  does  the  graph  show  an  exception  to  the  law  that 
the  volume  increases  with  a  rise  in  temperature  ? 


Temp. 

Volume 

Temp. 

Volume 

Temp. 

Volume 

0° 

1.000000 

6° 

.999914 

12° 

1.000334 

1 

.999948 

7 

.999962 

13 

1.000462 

2 

.999911 

8 

1.000003 

14 

1.000593 

3 

.999889 

9 

1.000068 

15 

1.000735 

4 

.999883 

10 

1.000147 

16 

1.000890 

5 

.999891 

11 

1.000239 

17 

1.001057 

202  APPLIED  MATHEMATICS 

Measurement  of  Heat 

105.  Units.  "When  a  definite  quantity  of  heat  is  applied  to 
various  substances  different  effects  are  produced,  depending  on 
the  nature  and  condition  of  the  substance.  An  amount  of  heat 
may  be  expressed  by  any  of  its  effects  which  can  be  measured ; 
but  it  has  been  found  convenient  to  measure  heat  by  considering 
the  change  in  temperature  it  produces. 

Two  heat  units  in  general  use  are  the  British  thermal  tmit 
(B.  t.  u.)  and  the  calorie.  For  ordinary  engineering  work  the 
unit  is  the  British  thermal  unit,  the  amount  of  heat  required 
to  raise  1  lb.  of  water  1°  F.  A  smaller  unit  used  in  laboratory 
investigations  is  the  calorie,  the  amount  of  heat  required  to 
raise  1  g.  of  water  1°  C.  The  amount  of  heat  required  to  raise 
a  quantity  of  water  1  degree  varies  with  the  temperatui-e ;  but 
the  variation  is  so  small  that  in  practical  work  we  may  neg- 
lect it  and  say  that  the  same  amount  of  heat  will  raise  1  lb.  of 
water  from  10°  to  11°  or  from  211°  to  212°. 

106.  The  relation  between  heat  and  work.  In  sawing 
wood,  boring  iron,  and  so  on,  a  part  of  the  energy  of  work 
becomes  heat.  It  has  been  found  possible  to  determine  the 
number  of  foot  pounds  of  work  required  to  raise  the  tempera- 
ture of  a  quantity  of  water  a  certain  number  of  degrees.  The 
famous  "experiments  of  Joule,  in  England,  in  the  years  1843~ 
1850,  showed  that  772  ft.  lb.  of  work  raised  the  temperature  of 
water  at  60°  F.,  1  degree. 

His  apparatus  consisted  of  a  brass  cylinder  in  which  water 
was  churned  by  a  brass  paddle  wheel,  which  was  made  to  re- 
volve by  a  falling  weight.  Later  experiments  by  other  methods 
have  given  results  more  nearly  exact,  and  by  general  consent 
it  is  now  considered  that  778  ft.  lb.  of  work  are  required  to 
raise  the  temperature  of  1  lb.  of  water  1**  F. 

1  B.  t.  u.  =  778  ft.  lb. 
1  ft.  lb.  =  .00129  B.  t.  u. 


HEAT  203 

PROBLEMS 

1.  How  many  British  thermal  units  are  required  to  raise 
the  temperature  of  120  lb.  of  water  from  the  freezing  point  to 
the  boiling  point? 

2.  On  a  cold  day  in  winter  a  tank  1  ft.  by  2  ft.  by  8  ft.  was 
filled  with  water  at  a  temperature  of  100°.  When  the  water  had 
reached  the  freezing  point,  how  much  heat  had  been  given  out  ? 

3.  If  1  lb.  of  coal  contains  13,500  B.  t.  u.  of  heat,  how  many 
pounds  of  coal  would  be  required  to  raise  the  temperature  of 
12  cu.  ft.  of  water  50°  if  there  was  an  efficiency  of  10  per  cent  ? 

4.  Find  the  number  of  British  thermal  units  required  to 
raise  the  temperature  of  20  lb.  of  lead  from  70°  to  the  melting 
point,  608°.  (It  takes  only  .03  as  much  heat  to  raise  1  lb.  of 
lead  1°  as  it  does  to  raise  1  lb.  of  water  1°.) 

5.  A  steel  ingot  weighing  1  T.  is  red-hot  (1200°).  How  much 
heat  is  given  off  as  it  cools  to  70°  ?  (It  takes  only  .12  as  much 
heat  to  raise  1  lb.  of  steel  1°  as  it  does  to  raise  1  lb.  of  water  1°.) 

6.  How  many  foot  pounds  of  work  are  required  to  raise  the 
temperature  of  20  lb.  of  water  12°  ? 

Solution.  778  x  20  x  12  =  186,720  ft.  lb. 

7.  The  temperature  of  1  cu.  ft.  of  water  was  raised  from  32° 
to  70°.    How  many  foot  pounds  of  work  did  it  require  ? 

8.  Through  how  many  feet  would  a  weight  of  1200  lb.  have 
to  fall  to  generate  enough  energy  to  raise  the  temperature  of 
8  lb.  of  water  15°  ? 

9.  Find  the  distance  through  which  a  weight  of  2  T.  could 
be  raised  by  the  expenditure  of  an  amount  of  heat  that  would 
raise  the  temperature  of  2  lb.  of  water  30°. 

10.  How  many  horse  power  would  it  take  to  raise  the  tem- 
perature of  10  cu.  ft.  of  water  from  70°  to  120°  in  1  hr.  ? 

62.4x10x778x50      .^ou 

Solution. — =  12.3  h.  p. 

60  X  33,000  ^ 


204  APPLIED  MATHEMATICS 

11.  Find  the  number  of  British  thermal  units  per  minute 
required  for  an  engine  of  the  following  dimensions :  diameter 
of  cylinder,  50  in. ;  stroke,  36  in. ;  revolutions  per  minute,  54 ; 
mean  effective  pressure,  28  lb.  per  square  inch.  Find  also  the 
number  of  pounds  of  coal  required  per  hour,  if  1  lb.  of  coal 
contains  13,500  B.  t.  u.  and  only  10  per  cent  of  the  heat  of  the 
coal  reaches  the  piston. 

12.  How  many  calories  are  required  to  raise  the  temperature 
of  40  g.  of  water  20°  C.  ? 

13.  If  126  calories  of  heat  raised  the  temperature  of  a 
quantity  of  water  49°  C,  how  many  grams  of  water  were  there  ? 

14.  The  temperature  of  1 1.  of  water  was  raised  from  40°  C. 
to  the  boiling  point.    How  many  calories  were  required  ? 

15.  How  many  calories  are  there  in  a  British  thermal  unit  ? 

16.  Construct  a  graph  to  change  calories  to  British  thermal 
units. 

Specific  Heat 

107.  Exercise.  Put  equal  weights  of  water  and  mercury  in 
similar  dishes.  Note  the  temperature  of  each.  Place  the  dishes 
on  an  electric  stove  or  in  a  dish  of  boiling  water.  After  a  time 
it  will  be  found  that  there  is  a  considerable  difference  in  the 
temperatures  of  the  mercury  and  water. 

Since  the  mercury  and  the  water  have  received  the  same 
amount  of  heat,  it  is  evident  that  it  takes  less  heat  to  raise  the 
temperature  of  1  lb.  of  mercury  1°  than  is  required  for  1  lb.  of 
water.  It  is  found  by  experiment  that  equal  weights  of  differ- 
ent substances  require  different  amounts  of  heat  to  raise  their 
temperatures  the  same  number  of  degrees.  Thus  1  lb.  of  water 
requires  1  B.  t.  u.  to  raise  its  temperature  1°  F. ;  1  lb.  of  glass 
requires  .2  B.  t.  u. ;  1  lb.  of  cast  iron  requires  .12  B.  t.  u. ;  and 
1  lb.  of  ice  requires  .5  B.  t.  u. 

108.  Definition.  The  specific  heat  of  a  substance  is  the  quo- 
tient obtained  by  dividing  the  amount  of  heat  required  to  raise 
the  temperatui'e  of  a  given  weight  of  it  1°  by  the  amount  of 


HEAT  205 

heat  required  to  raise  the  temperature  of  an  equal  amount 
of  water  1°.    (Note  the  similarity  to  specific  density.) 

The  specific  heat  of  substances  varies  a  little  with  the  tem- 
perature, but  in  practice  it  is  considered  to  be  constant. 

Table  of  Specific  Heat 

Aluminum 0.21  Silver 0.06 

Brass 0.09  Steel 0.12 

Copper 0.09  Tin 0.06 

Glass 0.20  Zinc 0.09 

Iron,  cast 0.12  Water 1.00 

Iron,  wrought      ....  0.11  Ice 0.50 

Lead 0.03  Steam 0.48 

Mercury 0.03  Air 0.24 

PROBLEMS 

1.  How  many  British  thermal  units  are  required  to  raise 
the  temperature  of  10  lb.  of  copper  from  70°  to  200°  ? 

Solution.  200°  -  70°  =  130°. 

It  would  require  1300  B.  t.  u.  to  raise  the  temperature  of  10  lb.  of 
water  130°.    Specific  heat  of  copper  =  .09. 

.-.  1300  X  .09  =  117  B.  t.  u. 

2.  How  many  calories  are  required  to  raise  the  temperature 
of  500  g.  of  lead  40°  C.  ? 

Solution.  500  x  40  x  .03  =  600  calories. 

3.  Find  the  amount  of  heat  required  to  raise  the  tempera- 
ture of  (a)  20  lb.  of  silver  from  70°  to  the  melting  point,  733°; 

(b)  30  lb.  of  aluminum  from  70°  to  the  melting  point,  1157° ; 

(c)  25  lb.  of  ice  from  0°  to  32°  ;  (d)  1  kg.  of  mercury  80°  C. 

4.  How  many  British  thermal  units  are  given  ofE  by  an  iron 
casting  which  weighs  50  lb.,  as  it  cools  from  2000°  to  70°  ? 

5.  If  1  lb.  of  water  at  70°  and  1  lb.  of  mercury  at  70°  are 
given  the  same  amount  of  heat,  how  hot  will  the  mercury 
become  when  the  water  is  at  73°  ? 


206  APPLIED  MATHEMATICS 

6.  Equal  weights  of  tin  and  cast  iron  are  put  into  a  tank 
of  boiling  water.  When  the  tin  has  been  heated  10°,  how  much 
has  the  iron  been  heated  ? 

7.  If  15  lb.  of  water  at  200°  and  8  lb.  of  water  at  70°  are 
mixed  together,  what  is  the  resulting  temperature  ? 

Solution.   Let  t  =  the  resulting  temperature. 

15  (200  —  t)  =  number  of  British  thermal  units  lost. 
8  (<  —  70)  =  number  of  British  thermal  units  gained. 

8  (<- 70)  =  15  (200-0- 
Solving,  t  =  154.8°. 

Check.   15  (200  -  154.8)  =  8  (154.8  -  70). 
15  X  45.2  =  8  X  84.8. 
678  =  678.4. 

8.  20  lb.  of  water  at  the  freezing  point  are  poured  into  25  lb. 
of  water  at  the  boiling  point.  What  is  the  temperature  of  the 
mixture  ? 

9.  A  tank  2  ft.  by  3  ft.  by  6  ft.  is  two  thirds  full  of  water 
at  60°.  If  the  tank  is  filled  with  water  at  120°,  what  is  the 
temperature  of  the  mixture  ? 

10.  How  many  pounds  of  water  at  40°  must  be  mixed  with 
60  lb.  of  water  at  100°  to  obtain  a  temperature  of  80°  ? 

Solution.   Let         p  =  number  of  pounds  at  40°. 

p  (80  —  40)  =  number  of  British  thermal  units  gained. 
60(100  —  80)  =  number  of  British  thermal. units  lost. 
;>  (80 -40)  =  60  (100 -80). 
p  =  30  lb. 
Check.      30  (80  -  40)  =  1200  B.  t.  u.  gained. 
60  (100  -  80)  =  1200  B.  t.  u.  lost. 

11.  How  many  pounds  of  water  at  180°  must  be  mixed  with 
1  cu.  ft.  of  water  at  56°  to  obtain  a  temperature  of  112°  ? 

12.  How  many  grams  of  water  at  0°  C.  must  be  mixed  with 
1  kg.  of  water  at  100°  C.  to  obtain  a  temperature  of  80°  C.  ? 

13.  How  much  water  at  212°  and  how  much  water  at  32° 
must  be  taken  to  make  up  36  lb.  at  97°  ? 


HEAT  207 

Solution.   Let  x  =  number  of  pounds  at  212°. 

y  =  number  of  pounds  at  32°. 

a;  +  y  =  36.  (1) 

X  (212  -  97)  =  2^  (97  -  32).  (2) 

115  x  =  65y.  (3) 

x  =  ^ly.  (4) 

Substitute  (4)  in   (1),  ^f  ,,  +  ^  =  36.  (5) 

y  =  23.  (6) 

Substitute  (6)  in  (1),  x  =  13.  (7) 

Check.  13  (212  -  97)  =  23  (97  -  32). 

13  X  115  =  23  X  65. 
115  =  115. 

14.  How  much  water  at  180°  and  at  81°  must  be  taken  to 
fill  a  tank  which  contains  90  lb.,  if  it  is  desired  to  have  the 
temperature  of  the  mixture  125°  ? 

15.  Into  a  dish  containing  some  water  at  4°C.  was  poured 
some  water  at  75°  C.  How  many  grams  of  each  were  taken  if 
there  were  in  all  250  g.  at  a  temperature  of  60°  C.  ? 

16.  An  iron  casting  when  red-hot  (1300°)  was  put  into  a  tank 
containing  2  cu.  ft.  of  water  at  170°.  If  the  temperature  of  the 
water  rose  to  200°,  what  was  the  weight  of  the  casting  ? 

Solution.   Let  x  =  number  of  pounds  of  cast  iron. 

Specific  heat  of  cast  iron  =  .12. 

2  cu.  ft.  of  water  =  62.4  x  2  =  124.8  lb.  of  water. 
(1300  —  200)  X  .12  X  =  number  of  British  thermal  units  lost 

by  the  iron. 
(200  -  170)  X  124.8  =  number    of    British    thermal    units 

gained  by  the  water. 
(1300  -  200)  X  .12  X  =  (200  -  170)  x  124.8. 
132  x  =  3744. 
X  =  28.4. 

Check.   28.4  x  1100  x  .12  =  3749  B.  t.  u.  lost. 

124.8  X  30  =  3744  B.  t.  u.  gained. 

17.  If  a  mass  of  lead  at  500°  was  put  in  a  gallon  of  water 
(8j  lb.)  and  the  temperature  of  the  water  rose  from  77°  to  80°, 
what  was  the  weight  of  the  lead  ? 


208 


APPLIED  MATHEMATICS 


18.  An  80-lb.  mass  of  steel  at  1000°  is  placed  in  a  tank  con- 
taining water  at  60°.  If  the  final  temperature  is  70°,  how 
many  pounds  of  water  are  in  the  tank  ? 

19.  If  20  lb.  of  brass  at  300°  were  placed  in  a  tank  containing 
1  cu.  ft.  of  water  at  72°,  what  would  be  the  final  temperature  ? 

20.  If  500  g.  of  brass  at  100°  C.  were  placed  in  188  g.  of 
water  at  17.5°  C.  and  the  final  temperature  was  33.5°  C,  find 
the  specific  heat  of  the  brass. 

Solution.    Let  s  =  the  specific,  heat  of  the  brass. 

500  (100  —  33.5)  s  =  number  of  calories  lost  by  the  brass. 
188(33.5  —  17.5)  =  number  of  calories  gained  by   the 

water. 
500(100  -  33.5)  s  =  188(33.5  -  17.5). 
188  X  16 
*  ~  500  X  66.5 
=  .0905. 
Check.   500  x  66.5  x  .0905  =  3010  calories  lost. 

188  X  16  =  3008  calories  gained. 


21.  The  following  data  were  obtained  by  experiment, 
the  specific  heat  of  each  metal. 


Find 


No. 

Water 

(grams) 

Tempera- 
ture of 
Water 

Material 

Weight 
used 

Initial 
Temperature 

Final  Tem- 
perature 

1 
2 
3 

188 
188 
188 

18.5° 
11.0° 
16.-5° 

Zinc 

Cast  iron 
Lead 

250  g. 
750  g. 
700  g. 

100°  C. 
100°  C. 
100°  c. 

28.5°  C. 
39.0°  C. 
26.0°  C. 

22.  45  g.  of  zinc  at  100°  C.  were  immersed  in  52  g.  of  water 
at  10°  C.  If  the  temperature  of  the  water  rose  to  17°  C,  find 
the  specific  heat  of  the  zinc,  assuming  that  no  heat  was  absorbed 
by  the  dish  containing  the  water. 

23.  A  room  20  ft.  by  30  ft.  by  10  ft.  is  to  be  heated  from  a 
temperature  of  32°  to  72°.  Assuming  that  1  cu.  ft.  of  air  at  32° 
weighs  .08  lb.,  that  the  specific  heat  of  air  is  .24,  and  that  8  per 


HEAT 


209 


cent  of  the  fuel  is  available  for  raising  the  temperature,  how 
many  pounds  of  hard  coal  (1  lb.  coal  =  13,500  B.  t.  u.)  would 
be  required  ? 

Latent  Heat 

109.  Exercise.  Place  a  dish  of  melting  ice  on  a  stove.  Though 
the  melting  ice  and  water  receive  heat  continuously,  a  ther- 
mometer placed  in  the  dish  will  stand  at  32°  F.  till  all  the  ice 
is  melted.  Then  the  mercury  will  rise  till  the  boiling  point  is 
reached.  The  temperature  will  remain  at  212°  till  all  the  water 
is  evaporated. 

110.  Latent  heat.  This  heat  which  goes  into  a  substance 
and  produces  a  change  in  form  rather  than  an  increase  in 
temperature  is  called  latent  (hidden)  heat. 

The  following  table  gives  the  approximate  number  of  British 
thermal  units  absorbed  by  1  lb.  of  the  substance  in  changing 
from  solid  to  liquid  or  liquid  to  solid. 


Latent  Heat  of  Fusion 

Bismuth 22.75 

Cast  iron 42.5 

Ice 142.6 

Lead 9.66 

Silver 43. 

Tin .     .  27. 

Zinc 54. 


Latent  Heat  of  Vaporization 


Alcohol 
Ether  . 
Mercury 
Water  . 
Water  . 
Water  . 


363  at  172°  F. 

162  at    93°  F. 

117  at  580°  F. 

965.7  at  212°  F. 

1044.4  at  100°  F. 

1091.7  at    32°  F. 


PROBLEMS 

1.  Find  the  number  of  British  thermal  units  required  to 
melt  the  following  masses  of  metal  after  they  have  been 
brought  to  the  melting  point :  (a)  120  lb.  of  iron ;  (b)  24  lb.  of 
lead ;  (c)  55  lb.  of  silver ;  {d)  40  lb.  of  tin. 

Solution,   (a)      42.5  x  120  =  5100  B.  t.  u. 

2.  How  much  heat  is  given  out  by  50  lb.  of  molten  zinc  as 
it  becomes  solid  ? 


210      •  APPLIED  MATHEMATICS 

3.  How  much  heat  is  required  to  melt  16  lb,  of  tin  at  70° 
if  its  melting  point  is  442°  ? 

Solution.    Specific  heat  of  tin  =  .06. 
442'=  -  70°  =  372°. 
16  X  372  X  .06  =  357  B.  t.  u.  to  raise  to  442°. 
16  X  27  =  432  B.  t.  u.  to  melt. 
789  B.  t.  u.,  total. 

4.  How  much  heat  is  required  to  melt  150  lb.  of  lead  at  70° 
if  its  melting  point  is  622°  ? 

5.  1  T.  of  molten  iron  at  2200°  cooled  to  70°.  How  much 
heat  was  given  off  if  the  melting  point  was  2000°  ? 

6.  A  cake  of  ice  weighing  50  lb.  is  at  0°.  How  many  British 
thermal  units  are  required  to  melt  it  and  bring  the  water  to 
the  boiling  point  ? 

7.  If  1  lb.  of  ice  at  32°  is  put  into  2  lb.  of  water  at  80°,  how 
much  of  the  ice  will  melt  ? 

Solution.  (80°  -  32°)  2  =  96  B.  t.  u.  available  to  melt  the  ice. 

142.6  =  number   of   British  thermal  units  re- 
quired to  melt  1  lb.  of  ice  at  32°. 
96 

.67  lb.  ice  melted. 


142.6 

Check.  142.6  x  .67  =  96. 

Y-  +  32  =  80. 

8.  How  much  boiling  water  will  be  required  to  melt  12  lb. 
of  ice  at  32°  ? 

9.  What  would  be  the  final  temperature  of  the  water  if  16  lb. 
of  ice  at  32°  were  put  into  40  lb.  of  boiling  water  ? 

Solution.   Let  t  —  the  final  temperature. 

142.6  X  16  =  number  of  British  thermal  units  to  melt 
the  ice. 
40  (212  —  t)  =  number  of  British  thermal  units  lost. 
16  (/  —  32)  +  142.6  X  16  =  number  of  British  thermal  units  gained. 

40  (212  -  0  =  16  (<  -  32)  +  142.6  X  16. 
Solve  and  check. 


HEAT  211 

10.  5  lb.  of  molten  lead  at  the  melting  point  610°  were  poured 
into  50  lb.  of  water  at  70°.  What  is  the  resulting  temperature  ? 

11.  1  lb.  of  lead  at  212°  is  placed  on  a  cake  of  ice  at  30°. 
How  much  ice  will  it  melt  ? 

12.  How  many  pounds  of  steam  at  212°  will  melt. 20  lb.  of 
ice  at  32°  ? 

13.  How  many  pounds  of  zinc  at  500°  must  be  added  to 
100  lb.  of  water  at  75°  to  heat  it  to  100°  ? 

14.  20  lb.  of  ice  at  0°  are  immersed  in  200  lb.  of  water  at 
200°.  What  is  the  temperature  of  the  water  when  the  ice  has 
just  melted  ? 

15.  How  many  pounds  of  water  at  70°  would  be  evaporated 
at  212°  by  1  T.  of  coal,  assuming  an  efficiency  of  12  per  cent, 
and  13,500  B.  t.  u.  per  pound  of  coal  ? 

16.  The  temperature  of  1  lb.  of  water  in  a  teakettle  rises 
from  32°  to  212°  in  ten  minutes,  (a)  How  long  before  the 
kettle  will  boil  dry  ?  (b)  If  the  kettle  contained  5  lb.  of  water, 
how  many  British  thermal  units  would  be  needed  to  boil  it  dry  ? 

Solution,   (a)    212°  -  32°  =  180°. 

=  18  B.  t.  u.  per  minute. 

10  ^ 

965.7      .„_     .      , 

=  53.7  minutes. 

18 

17.  If  1  lb.  of  ice  at  0°  is  put  on  an  electric  stove  which  gives 
out  8  B.  t.  u.  per  minute,  find  the  number  of  British  thermal 
units  and  the  number  of  minutes  required  (a)  to  raise  the  ice 
to  32° ;  (p)  to  melt  the  ice ;  (c)  to  raise  the  water  to  212° ; 
(d)  to  evaporate  the  water ;  (e)  to  raise  the  steam  to  312°.  Con- 
struct a  graph  and  write  the  results  on  it.  Units :  horizontal, 
1  large  square  =  10  minutes ;  vertical,  1  large  square  =  20°. 


CHAPTER   XVII 


ELECTRICITY 


111.  Exercise.  Into  a  tumbler  two  thirds  full  of  water  pour 
2  ccm.  of  sulphuric  acid.  Stand  in  this  solution  a  strip  of  zinc 
and  a  strip  of  copper  each  well  sandpapered.  Take  6  ft.  of  No.  20 
insulated  copper  wire  and  wind  about  25  turns  around  a  large 
lead  pencil,  leaving  about  a  foot  uncoiled 
at  each  end.  Cut  the  insulation  from  the 
ends  of  the  wire  and  wrap  the  ends 
around  the  strips,  as  shown  in  Fig.  82. 
To  get  good  connections  it  may  be  neces- 
sary to  cut  into  the  edge  of  the  stri})s 
and  wedge  the  wire  under  the  pieces 
lifted. 

Take  a  piece  of  soft  wrought  iron  and 
sprinkle  some  iron  filings  on  each  end. 
Result  ?  Place  the  iron  within  the  coil, 
as  shown  in  Fig.  82,  and  drop  some  iron 
filings  on  the  ends.    Result  ?    Is  the  iron 

magnetized  ?  If  so,  we  have  generated  a  current  of  electricity  and 
magnetized  the  iron.  (See  Shepardson's  "Electrical  Catechism.") 

112.  Nature  of  electricity.  The  exact  nature  of  electricity 
is  not  known.  Some  scientists  think  it  is  a  condition  of  the 
ether.  Others  think  that  it  is  a  form  of  energy  or  force.  How- 
ever, much  is  known  about  the  laws  of  electricity  and  about 
methods  of  applying  it  to  useful  work. 

113.  Electromotive  force.  When  the  strips  of  copper  and 
zinc  were  placed  in  the  solution  of  sulphuric  acid,  the  acid  dis- 
solved the  zinc  strip  faster  than  it  did  the  copper  strip.    We 

212 


Fkj.  82 


ELECTRICITY  213 

say  that  this  caused  an  electrical  flow  from  the  zinc  to  the 
copper ;  that  is,  an  electromotive  force  exists  between  the  two 
strips.  Whatever  produces  or  tends  to  produce  an  electrical 
flow  is  called  an  electromotive  force  (e.  m.  f.).  When  the  two 
strips  are  connected  by  the  wire  this  action  takes  place  con- 
tinuously and  there  is  said  to  be  a  flow  of  electricity  from  the 
zinc  to  the  copper  and  through  the  wire  to  the  zinc  again. 
Though  we  cannot  perceive  this  flow  by  any  of  our  senses,  we 
can  see  the  effects  it  produces. 

114.  The  electrical  units.  It  is  not  possible  nor  is  it  neces- 
sary to  give  exact  definitions  here.  However,  definitions  can 
be  given  which  are  readily  understood  and  are  sufficiently  exact 
for  practical  purposes. 

The  volt.  We  may  think  of  the  electromotive  force  existing 
between  the  strips  of  zinc  and  copper  in  the  cell  described 
above,  as  pressure.  It  takes  pressure  to  force  a  current  of  elec- 
tricity through  a  wire,  just  as  it  takes  pressure  to  drive  a  stream 
of  water  through  a  pipe.  To  measure  this  pressure  we  have 
the  unit  of  electromotive  force  called  a  volt  (from  Volta,  an 
Italian  physicist  who  lived  from  1745  to  1827).  The  pressure 
of  a  gravity  or  crowfoot  cell  is  about  1.1  volts.  When  a  wire 
is  moved  across  the  magnetic  lines  of  force  which  exist  between 
the  poles  of  a  magnet,  an  electrical  flow  is  produced  in  the 
wire.  A  volt  is  the  electromotive  force  set  up  in  a  wire  that 
crosses  magnetic  lines  of  force  at  the  rate  of  one  hundred  mil- 
lion per  second.  In  a  dynamo  the  armature  may  be  thought  of 
as  a  bundle  of  wires  which  cut  across  the  lines  of  force  of  the 
field  magnet  as  the  armature  revolves. 

The  ohm.  The  pressure  (electromotive  force)  produces  a 
floAv  of  electricity  which  meets  with  resistance  in  the  conductor. 
Just  as  the  frictional  resistance  in  a  water  pipe  opposes  the 
flow  of  water,  so  the  electrical  resistance  of  a  conductor  opposes 
the  flow  of  electricity.  The  unit  of  resistance  is  the  ohm  (from 
Ohm,  a  German  mathematician  who  lived  from  1789  to  1854). 


214  APPLIED  MATHEMATICS 

The  olim  is  nearly  equal  to  the  resistance  of  1000  ft.  of  copper 
wire  .1  in.  in  diameter.  Different  substances  have  different  de- 
grees of  resistance.  The  resistance  of  metals  increases  slightly 
as  the  temperature  rises,  but  the  resistance  of  carbon  (incandes- 
cent lamp  filament)  decreases  with  a  rise  in  temperature.  Thus 
the  resistance  of  a  16  candle  power  incandescent-lamp  carbon 
filament  is  about  220  ohms  when  hot,  but  it  may  be  as  high  as 
400  ohms  when  cold. 

Resistance  variefe  directly  as  the  length  and  inversely  as  the 
cross  section  of  a  conductor.  Thus  if  the  resistance  of  100  ft. 
of  wire  is  2  ohms,  the  resistance  of  300  ft.  of  the  same  wire 
is  6  ohms ;  if  the  resistance  of  a  wire  .3  in.  in  diameter  (cross 
section,  7.07  sq.  in.)  is  8  ohms,  the  resistance  of  a  wire  of  the 
same  material  and  length  .6  in.  in  diameter  (cross  section, 
28.27  sq.  in.)  is  2  ohms. 

The  ampere.  The  unit  for  measuring  the  rate  of  the  electri- 
cal flow  is  the  ampere.  An  ampere  (from  Ampere,  a  French 
physicist  who  lived  from  1775  to  1836)  may  be  defined  as  the 
current  which  an  electromotive  force  of  1  volt  will  send  through 
a  conductor  whose  resistance  is  1  ohm. 

The  number  of  amperes  of  current  corresponds  quite  closely 
to  the  rate  of  flow  of  a  stream  of  water.  We  may  say  that  at  a 
certain  point  in  an  electrical  circuit  the  rate  of  flow  is  5  amperes, 
just  as  we  would  say  that  at  a  certain  point  in  a  water  pipe  the 
rate  of  flow  of  the  water  is  10  gal.  per  minute. 

Given  an  electromotive  force  of  1  volt,  a  circuit  of  1  ohm 
resistance,  and  we  have  a  current  of  1  ampere. 

115.  Ohm's  law.  A  very  simple  relation  exists  between  the 
electromotive  force,  resistance,  and  current  in  a  closed  circuit. 

Let  V  =  the  number  of  volts  of  electromotive  force, 
R  =  the  number  of  ohms  of  resistance, 
A  =  the  number  of  amperes  of  current, 

V 
and  we  have  Ohm's  law,  —  =  A. 
R 


ELECTRICITY  216 

In  words  this  law  may  be  stated  as  follows :  The  number  of 
volts  of  electromotive  force  divided  by  the  number  of  ohms 
of  resistance  gives  the  number  of  amperes  of  current  flow- 
ing through  a  circuit.  This  law  was  first  formulated  by  Ohm 
in  1827. 

PROBLEMS 

1.  How  many  amperes  are  there  in  a  circuit  of  20  ohms 
resistance  if  the  dynamo  generates  110  volts  ? 

c  F      110      ._ 

bOLUTiON.  —  = =  5.5  amperes. 

R        20  ^ 

2.  A  battery  sends  a  current  of  5  amperes  through  a  circuit. 
If  the  electromotive  force  is  10  volts,  find  the  total  resistance 
of  the  circuit.       y  <h'^'^ 

3.  If  a  cable  has  a  resistance  of  .004  ohm  and  a  current  of 

20  amperes  passes  through  it,  what  is  the  electromotive  force  ?  ,0°  "^ 

4.  Find  the  resistance  of  an  incandescent  lamp  which  takes 

a  current  of  .5  ampere  when  connected  to  a  110-volt  main.    I.'*-'*  ^ 

5.  If  a  telegraph  wire  has  a  resistance  of  200  ohms,  how    ^^ 
many  amperes  will  be  sent  through  it  by  a  pressure  of  10  volts  ?  * 

6.  The  wires  in  an  electric  heater  will  stand  8  amperes 
without  becoming  unduly  heated.  What  must  be  the  resistance 
for  110  volts  ?     y-S  rt  5  -'"^•^ " 

7.  A  dynamo  generates  110  volts.  What  is  the  total  resist- 
ance of  the  circuit  if  there  is  a  current  of  40  amperes  ?  i^^^  Ov*'-o 

8.  A  32  candle  power  lamp  for  a  220-volt  circuit  has  a 
resistance  of  330  ohms,  and  a  16  candle  power  lamp  for  a 
110-volt  circuit  has  a  resistance  of  180  ohms.  Which  lamp  has 
the  greater  current  ?    ^  tAifc. 

9.  Construct  a  curve  to  show  the  relation  between  the 
electromotive  force  and  the  resistance  of  a  circuit  in  which 
the  current  is  20  amperes,  as  the  resistance  varies  from  1  to 
10  ohms. 


216  APPLIED  MATHEMATICS 

10.  If  the  electromotive  force  of  a  dynamo  remains  constantly 
at  120  volts,  construct  a  curve  to  show  the  changes  in  the 
current  as  the  resistance  increases  from  10  to  120  ohms. 

116.  Resistances  in  combination.  In  the  preceding  problems 
the  resistance  of  the  circuit  was  considered  as  a  single  resist- 
ance, but  in  practical  work  the  circuit  is  made  up  of  several 
parts.  Thus  in  an  electric  lighting  system  the  total  resistance 
is  made  up  of  the  resistances  of  the  dynamo,  lamps,  and  con- 
necting wires.  The  parts  of  a  circuit  may  be  combined  in  two 
distinct  ways. 

117.  Series  circuits.  When  the  different  parts  of  a  circuit 
are  joined  end  to  end  and  the  whole  current  flows  through  each 
part,  the  circuit  is  called  a  series  cir- 
cuit. Let  D  in  Fig.  83  be  a  dynamo 
maintaining  an  electromotive  force  of 
110  volts  measured  across  the  termi- 
nals AB.  This  means  that  110  volts  ^  j-j^  §3 
are  continuously  generated  and  used 

up  in  forcing  the  current  through  the  circuit  BCEFA .  Hence  we 
may  say  that  from  Bto  A  there  is  a  drop  in  voltage  of  110  volts. 
Let  C,  E,  and  F  be  arc  lights  having  resistances  of  4.2,  4.6, 
and  4.8  ohms  respectively,  and  let  the  resistance  of  the  line 
be  4  ohms. 

Total  resistance  =  4.2  -f  4.6  -f-  4.8  -f-  4 
=  17.6  ohms. 

By  Ohm's  law,  —  =  z-=-x  =  6.25  amperes. 

At  any  point  in  the  circuit  the  current  is  6.25  amperes,  since 
in  a  series  circuit  the  current  is  constant.  But  there  is  a  con- 
tinualdrop  in  the  voltage  along  the  circuit  as  the  voltage  is 
used  up  in  forcing  the  current  along  its  path.  This  drop  in 
voltage,  or  drop  of  potential,  as  it  is  sometimes  called,  follows 
Ohm's  law. 


ELECTRICITY  217 

The  drop  in  lamp  C=AR  =  6.25  x  4.2  =  26.2  volts. 
The  drop  in  lamp  E  =  A  ■  It  =  6.25  x  4.6  =  28.8  volts. 
The  drop  in  lamp  F  =  AR  =  6.25  x  4.8  =  30.0  volts. 
The  drop  iij  the  line  =  AR  =  6.25  x  4     =    25.0  volts. 

Total  drop  =  110.0  volts.   Check. 

The  arc  lights  in  general  use  to  light  city  streets  are  connected 
in  series,  and  the  entire  current  goes  through  each  lamp. 

118.  Ammeter.  The  number  of  amperes  of  current  is  meas- 
ured by  an  ammeter.    It  consists  of  a  coil  of  wire  suspended 


Fig,  84 

between  the  poles  of  a  magnet  so  that  it  rotates  through  a 
small  angle  when  the  current  passes  through.  The  coil  carries 
a  light  needle.  The  instrument  is  graduated  by  passing  through 
it  currents  of  known  strength,  and  marking  on  the  scale  the 
position  of  the  needle.  The  type  of  ammeter  commonly  used 
is  cut  into  the  circuit  when  a  measurement  is  made. 

119.  Voltmeter.  Voltage  (electromotive  force,  drop  of  poten- 
tial) is  measured  by  the  voltmeter.  Most  voltmeters  are  simply 
special  forms  of  ammeters.  The  voltmeter  also  is  graduated 
by  experiment.    It  is  put  on  circuits  of  known  voltage  and  the 


218 


APPLIED  MATHEMATICS 


position  of  the  needle  is  marked  on  the  scale.  In  using  the 
voltmeter  its  terminals  are  connected  to  the  ends  of  the  parts 
of  the  circuit  in  which  the  voltage  is  to  be  measured;   the 


Fig.  86 


reading  of  the  voltmeter  is  the  number  of  volts  of  electro- 
motive force,  or  drop  of  potential.  If  a  voltmeter  is  connected 
across  the  terminals  of  an  arc  light  and  the  reading  is  47  volts, 
it  means  that  47  volts  are  used  up  in  running  that  arc  light. 

In  Fig.  86  the  ammeter  A  is  arranged  to  measure  the  current 
produced  by  the  dynamo  D ;  and  the  voltmeter  V  is  connected 


Fig.  87 


to  show  the  electromotive  force  between  the  terminals  of  the 
dynamo.  Fig.  87  shows  an  ammeter  and  a  voltmeter  arranged 
to  measure  the  current  and  drop  in  voltage  in  an  arc  lamp  L. 


ELECTRICITY  219 


PROBLEMS 


1.  Three  wires  having  resistances  of  2,  5,  and  8  ohms 
respectively  are  joined  end  to  end  and  a  voltage  of  90  volts  is 
applied.    How  many  amperes  of  current  are  there  ?       (p  o^^-^  • 

2.  Two  wires  of  resistances  6  and  8  ohms  respectively  are  \  0.: 
joined  in  series.    If  the  current  is  1.8  amperes,  find  the  voltage._l!lL 

3.  Two  incandescent  lamps  are  in  series  and  one  has  twice       ' 
as  great  resistance  as  the  other.    If  the  voltage  is  110  and  the 
current  is  ^  of  an  ampere,  find  the  resistance  of  each  lamp. 

Solution.   Let  K  =  the  resistance  of  one  lamp. 

2  R  =  the  resistance  of  the  other  lamp. 

F  ^  110  _  1 

R      SR      S' 

R  =  110  ohms. 

2R  =  220  ohms. 

Total  =  330  ohms. 

R      330      3 

4.  Find  the  internal  resistance  of  a  battery  which  gives  a 
current  of  1.5  ampeifes  with  an  electromotive  force  of  5  volts, 
if  the  external  resistance  is  1.33  ohms. 

5.  An  iron  wire  and  a  copper  wire  are  in  series.  If  the 
voltage  is  12  volts,  the  current  2.8  amperes,  and  the  copper  wire 
has  a  resistance  of  3  ohms,  find  the  resistance  of  the  iron  wire. 

6.  A  circuit  consists  of  two  wires  in  series.  An  electro- 
motive force  of  30  volts  gives  a  current  of  3.2  amperes.  If  the 
length  of  one  wire  is  doubled  and  the  other  is  made  5  times 
as  long,  the  current  is  .84  ampere.  Find  the  resistance  of  the 
two  wires. 

7.  What  voltage  is  necessary  to  furnish  a  current  of  9.6 
amperes,  if  the  circuit  is  made  up  of  2  mi.  of  No.  6  Brown  & 
Sharpe  gauge  copper  wire  (resistance  of  1000  ft.,  .395  ohm)  and 
10  arc  lights  in  series,  each  of  4.8  ohms  resistance  ?  Find  also 
the  drop  in  voltage  in  the  wire  and  in  the  lamps. 


220  APPLIED  MATHEMATICS 

8.  A  dry  cell  is  used  to  ring  a  door  bell.  The  resistance  of 
the  wire  in  the  bell  is  1.5  ohms,  of  the  line  .5  ohm,  and  of  the 
cell  1  ohm.  If  the  electromotive  force  of  the  cell  is  1.4  volts, 
what  current  will  flow  when  the  circuit  is  closed  ? 

9.  What  is  the  resistance  per  mile  of  No.  20  Brown  &  Sharpe 
gauge  copper  wire,  if  the  voltmeter  connected  to  the  ends  of 
100  ft.  of  the  wire  reads  5.13  volts  and  the  ammeter  reads 
5  amperes  ? 

10.  An  arc-light  dynamo  of  30  ohms  resistance  supplies  a 
current  of  6.8  amperes  through  12  mi.  of  No.  6  Brown  &  Sharpe 
gauge  copper  wire  to  a  series  of  50  arc  lights,  each  adjusted  to 
47  volts.    Find  the  electromotive  force  of  the  dynamo. 

Suggestion.  The  drop  in  voltage  in  the  lamps  =  47  X  50.  Find 
the  drop  iu  voltage  in  the  dynamo  and  in  the  line  by  V  =  R  -A.  The 
total  voltage  is  the  sum  of  the  drop  in  voltage  in  the  three  parts  of 
the  circuit.  Check  by  finding  the  total  resistance  of  the  circuit  and 
dividing  it  into  the  total  electromotive  force ;  this  should  give  6.8 
amperes. 

11.  In  an  electric  lighting  system  there  are  6  mi.  of  No.  6 
Brown  &  Sharpe  gauge  copper  wire,  and  80  arc  lights,  each 
having  a  resistance  of  4.5  ohms.  The  resistance  of  the  dynamo 
is  3  ohms  and  the  electromotive  force  is  3725  volts.  Find 
(a)  the  total  resistance ;  (h)  the  current ;  (c)  the  fall  of  voltage 
in  the  dynamo,  line,  and  lamps. 

12.  The  voltage  across  the  mains  of  an  electric-light  circuit 
is  110  volts.  If  a  voltmeter  is  connected  across  the  mains  in 
series  with  a  resistance  of  6000  ohms,  it  reads  70  volts.  What 
is  the  resistance  of  the  voltmeter  ? 

Solution.    Since  there  is  a  drop  of  70  volts  in  the  voltmeter, 
there  is  a  drop  of  110  —  70  =  40  volts  in 
the  resistance. 

F_  J0_         1 

R  ~  UOOO  "  150  '^^P^'"*^-  Pj^^  gg 


ELECTRICITY  221 

Since  the  current  is  the  same  in  all  jmrts  of  the  circuit,  —  = = 

10,500  ohms,  resistance  of  the  voltmeter.  

150 
Check.  10,500  +  0000  =  16,500  ohms. 

.   V        110  1 

ie  =  10:500  =  150  ""^P^^^- 

13.  A  coil  of  wire  is  placed  in  series  with  a  voltmeter  having 
a  resistance  of  18,000  ohms  across  110-volt  mains.  If  the  volt- 
meter reading  is  60  volts,  find  the  resistance,  of  the  coil  of  wire. 

14.  A  voltmeter  has  a  resistance  of  10,000  ohms.  What 
will  be  the  reading  of  the  voltmeter  when  connected  across 
110-volt  mains  with  a  man  having  a  hand-to-hand  resistance 
of  5000  ohms  ? 

120.  Multiple  circuits.    When  the  branches  of  a  circuit  are 
connected  so  that  only  a  part  of  the  current  flows  through  each 
of  the  several  branches,  the  circuit  is     . 
called  a  mtdtlple,  parallel,   or  divided 
circuit.    Fig.  89  shows  three  incandes-  •* 
cent  lamps  connected  in  multiple.    The  fig.  89 
ordinary   incandescent    lamps   used   in 

houses  are  connected  in  multiple  between  mains  from  the 
terminals  of  the  dynamo.  The  full  electromotive  force  of  the 
dynamo,  except  the  drop  in  voltage  in  the  wires,  acts  upon  each 
lamp ;  but  only  a  part  of  the  current  goes  through  each  lamp. 

121.  To  find  the  total  resistance  of  a  multiple  circuit.  In 
Fig.  90  let  the  drop  in  voltage  from  £  to  C  be  12  volts,  and 
a  and  h  have  resistances  of  2  and  4  ohms  respectively.  The 
pressure  (electromotive  force)  in  each  branch  is  12  volts ;  just 
as  in  a  water  pipe  of  similar  construction  the  pressure  would 
be  the  same  in  each  branch. 


ST5 


F      12      ^ 

—  =  -—  =  {i  amperes  m  a.  — ^ 

R  L  A  E 


12 


Z  ohms 


A  ohma 


-r-  =  3  amperes  in  h.  ^ 

4  Fig.  90 


222 


APPLIED  MATHEMATICS 


Hence  the  total  current  is  6  +  3  =  9  amperes. 
The  total  resistance  of  a  and  h  is  given  by 


R  = 


12      4    ^ 
-  =  -ohms. 


We  will  now  obtain  a  general  formula  for  the  total  resistance 
of  a  multiple  circuit. 

**       Let  V  =  the  drop  in  voltage  from  B  to  C. 

r^  =  resistance  of  a. 
7*2  =  resistance  of  b. 
R  =  total  resistance. 

—  =  current  in  a.  ° 
r^                                                                 Fig.  91 

—  =  current  in  b. 

T 

V       V       V(r^  +  r^      ^  ^  , 

1-  —  =  — ^ =  total  current. 


Q 

7f  ohma 

, 

A          B 

C           0 

^  onms 

But 


or 


—  =  total  current. 
R 

R  i\r^ 

R  = 


1  T^   i 


(1) 


In  a  multiple  circuit  of  two  branches  the  total  resistance  is  the 
product  of  the  resistances  divided  by  their  sum. 

In  a  similar  manner  let  the  student  work  out  the  formulas 
for  three  and  four  branches,  obtaining : 


R.= 


riVs 


R. 


^*'2  +  'Ys  +  'Yz 


r  r  r  A-  r  v  r  A-  r  r  r  A-  r  r  r 

1    2   8     1^      2   3   4'        341'        412 


(2) 

(3) 


ELECTRICITY 


223 


When  two,  three  or  four  equal  resistances  are  combined  in 
multiple,  we  have  from  (1),  total  resistance  =  j^  =  7; ' 


3' 


3r2 
r*    _r 

Thus  when  ten  16  candle  power  lamps  of  220  ohms  resistance 
are  connected  in  multiple  the  total  resistance  is 
220 


from  (2),  total  resistance 
from  (3),  total  resistance 


10 


=  22  ohms. 


122.  Graphical  method  of  finding  the  total  resistance.   The 

total  resistance  of  a  multiple  circuit  can  be  readily  determined 
by  a  graph. 

EXERCISES 

1.  Construct  a  graph  to  find  the  result  of  combining  resist- 
ances of  20  and  30  ohms  in  midtiple. 

Take  OX  any  convenient  length,  and  with  convenient  units  lay  ofE 
OM  =  30  ohms,    and   XN  =  20 
ohms.    Draw  ON  and  XAI,  inter- 
secting at  ^.   ^5  =  12  ohms,  the 
total  resistance. 

OMXN 


That  is,  AB  = 


(1) 


OM+XN 

Prove  geometrically.  The  two 
pairs  of  similar  triangles  OB  A, 
OXN ;  &TidiXBA,X  OM  give  two 
equations.  Eliminating  XB  from 
these  equations  gives  (1).  (See 
Problem  14,  p.  77.) 

A  similar  construction  gives  -^^^  92 

the    total    resistance   of    any 

number  of  resistances  in  multiple.    Thus,  given  the  resistances 
20,  30,  and  18  ohms,  combine  20  and  30  ohms,  as  above.   Then 


Y 

30 
25 
SO 
15 
10 
5 
0 

r^ 

\ 

\ 

s. 

N 

\ 

\ 

^ 

P 

\ 

K 

/ 

^ 

y' 

s 

% 

^ 

y 

/ 

/ 

1 
1 

1 

\ 

< 

D 

&         D            X 

224 


APPLIED  MATHEMATICS 


lay  off  XP  =  18  ohms.  Draw 
PB,  intersecting  XA  at  C,  and 
CD  is  the  total  resistance. 

2.  What  resistance  must  be 
combined  with  24  ohms  to 
obtain  a  total  resistance  of 
8  ohms? 

Take  OX  any  convenient 
length  and  with  a  convenient  unit 
lay  off  OM  =  24  ohms.  Draw 
MX.  On  MX  take  a  point  A, 
such  that  ^JB  =  8ohms.  Draw 
OA,  and  extend  to  meet  XP  at 
N.   XN  =  12  ohms,  the  required  resistance. 

The  graphical  method  should  be  used  to  solve  and  check  some  of 
the  following  problems. 

PROBLEMS 

1.  Three  resistances  of  20,  30,  and 

40  ohms  are  joined  in  multiple.    Find 

the  total  resistance. 

20.30-40 


Y 

30 
Z5 
ZO 
15 
lO 
5 
O 

P 

^, 

^ 

N, 

X 

\ 

^ 

N 

^ 

-^ 

N, 

^- 

^ 

^ 

j 

V 

\ 

0                                              B                      X 

Fig.  93 


Solution.  R  = 


ZOohms 


40ohms 


20-30 +  20-40 +  30-40 
9.2  ohms. 


Fig.  94 


Solution. 


2.  If  110  volts    be   applied  to  the   circuit   in  Problem  1, 
what  is  the  total  current  and  the  current  in  each  branch  ? 
V  _  110 
R  ~  9.2 

5.5  amperes. 

3.7  amperes. 

Jj^-  =  2.8  amperes. 

12  amperes. 


110 
'Ho' 
1  10 


12  amperes. 


Check. 


3.  Three  lamps  having  resistances  of  60,  120,  and  240  ohms 
are  connected  in  multiple.  If  they  are  supplied  with  110  volts, 
find  the  total  resistance,  the  total  current,  and  the  current  in 
each  lamp. 


ELECTRICITY 


225 


4.  Two  lamps  of  100  and  150  ohms  are  put  in  parallel  with 
each  other,  and  the  pair  is  joined  in  series  with  a  lamp  of 
100  ohms.  If  the  electromotive  force  is  200  volts,  what  will 
be  the  current? 

5.  A  resistance  of  10  ohms  is  put  in  parallel  with  an  un- 
known resistance.  If  an  electromotive  force  of  120  volts  gives 
a  current  of  20  amperes,  find  the  unknown  resistance. 

Solution.   Let  r  =  the  unknown  resistance. 

10  r 


Then 


10  +  r 
120 


=  the  total  resistance. 


20 


=  G  =  the  total  resistance. 


Check. 


-1^  =  6. 
10 +  r 

10  r  =  60  +  6  r. 

r  =  15  ohms 

10  X  15      150 


10  +  15       25 


=  6  ohms. 


6.  A  lamp  of  unknown  resistance  is  put  in  parallel  with  a 
lamp  of  220  ohms  resistance.  If  a  voltage  of  110  volts  gives 
a  current  of  1.6  amperes,  what  is  the  unknown  resistance  ? 

7.  The  total  resistance  of  three  wires  in  multiple  is  1.52 
ohms.  If  the  resistance  of  two  of  the  wires  is  3  and  5  ohms 
respectively,  what  is  the  resistance  of  the  third  ? 

8.  The  total  resistance  of  two  conductors  in  multiple  is 
4.8  ohms,  and  the  sum  of  the  two  resistances  is  20 ;  find  them. 


Fig.  95 

9.  The   total   resistance   between  A    and  B  in   Fig.  95  is 
6.25  ohms.    Find  the  resistance  x. 


226  APPLIED  MATHEMATICS 

10.  Three  resistances  in  parallel  are  in  the  ratio  1:2:3. 
If  an  electromotive  force  of  120  volts  gives  a  current  of 
11  amperes,  find  each  resistance. 

11.  Twenty  16  candle  power  110-volt  lamps  are  in  multiple. 
If  the  resistance  of  each  lamp  is  220  ohms,  what  is  the  total 
resistance,  and  what  is  the  current  ? 

12.  A  110-volt  incandescent  lighting  circuit  divides  into 
three  multiple  circuits  of  5,  8,  and  10  lamps  respectively.  If 
the  resistance  of  each  lamp  is  220  ohms,  find  (a)  the  resistance 
of  each  branch  ;  (b)  the  total  resistance ;  (c)  the  current ;  (cl)  the 
current  in  each  branch. 

13.  Construct  a  curve  to  show  the  change  in  the  resistance 
of  a  multiple  circuit  consisting  of  a  number  of  incandescent 
lamps  of  220  ohms,  as  the  number  of  lamps  increases  from 
1  to  20. 

14.  Two  conductors  of  12  and  18  ohms  respectively  are  in 
multiple.  What  resistance  must  be  placed  in  series  with  the 
multiple  circuit  to  give  a  current  of  5  amperes  with  an  electro- 
motive force  of  110  volts  ? 

Work  and  Power 

123.  The  watt.  When  an  electromotive  force  overcomes  the 
resistance  of  a  conductor  and  causes  a  current  to  flow,  work  is 
done.  This  is  analogous  to  the  case  where  work  is  done  by  the 
pressure  of  steam  on  the  piston  of  an  engine.  The  number  of 
pounds  pressure  multiplied  by  the  number  of  feet  through 
which  the  piston  is  moved  gives  the  number  of  foot  poimds  of 
work.  Power  is  the  rate  of  doing  work.  The  unit  of  mechanical 
power  is  the  horse  power,  the  rate  of  doing  work  equal  to 
33,000  ft.  lb.  per  minute.  The  unit  of  electrical  power  is  the 
watt  (James  Watt,  Scotland,  1736-1819,  practically  the  in- 
ventor of  the  modern  steam  engine),  the  rate  of  doing  work 
equal  to  44^  ft.  lb.  per  minute. 


ELECTRICITY  227 

Power  in  watts  equals  the  number  of  volts  multiplied  by  the 
number  of  amperes. 

W=V-A.  (1) 

Thus  if  a  dynamo  supplies  a  current  of  50  amperes  at  a  voltage 
of  110  volts,  the  power  delivered  is  110  x  50  =  5500  watts. 

V 
From  Ohm's  law  —  =  A,  (1)  may  be  written 

W  =  A''R.  (2) 

W  =  --  (3) 

In  words  :  watts  equal  volts  mtdtiplied  by  arnperes ;  (1) 

watts  equal  current  squared  Tnultiplied  by  resistance ;  (2) 

watts  equal  volts  squared  divided  by  resistance.  (3) 

To  express  watts  in  horse  power : 

Since  1  h.  p.  =  33,000  ft.  lb.  per  minute, 

and  1  watt  =  44i  ft.  lb.  per  minute, 

.,  33,000      ,. 

1  h.  p.  =  —h- —  watts. 

1  h.  p.  =  746  watts. 

124.  The  kilowatt.  For  many  purposes  a  larger  unit  than 
a  watt  is  convenient.  Hence  1000  watts,  called  a  kilowatt  (kw.), 
is  sometimes  taken  as  the  unit  of  power. 

125.  The  kilowatt  hour.  A  kilowatt  hour  is  a  practical  unit 
used  in  measuring  electrical  energy.  It  is  the  energy  expended 
by  1  kw.  in  1  hr.  Thus  20  kw.  hr.  might  mean  2  kw.  for  10  hr., 
5  kw.  for  4  hr.,  1  kw.  for  20  hr.,  and  so  on. 

1  kw.  hr.  =  441  X  1000  x  60  ft.  lb. 

1  h.  p.  hr.  =  33,000  x  60  ft.  lb. 

^  ...  441  X  1000x60 ,        , 

Hence  1  kw.  hr.  = -33^^^^^^-^^  h.  p.  hr. 

1  kw.  hr.  =  1.34  h.  p.  hr. 


228  APPLIED  MATHEMATICS 

PROBLEMS 

1.  An  arc  light  requires  10  amperes  at  45  volts.  How  much 
power  does  it  absorb  ? 

Solution.         IT  =  F-vl  =  45  x  10  =  450  watts. 

450       „, 
746  =  -6h.p. 

2.  A  16  candle  power  incandescent  lamp  is  on  a  110-volt 
circuit  and  takes  ^  ampere.  How  many  watts  per  candle 
power  are  required  ? 

Solution.  TF  =  F-yl  =  110   i  =  55  watts. 

55 

-^  =  3.5  watts  per  candle  power. 

3.  A  dynamo  has  a  voltage  of  550  volts  and  is  producing 
40  kw.    How  many  amperes  in  the  current  ? 

4.  How  many  watts  will  be  lost  in  forcing  the  current 
through  the  armature  of  a  dynamo,  if  the  resistance  is  .035  ohm 
and  the  current  is  30  amperes  ? 

5.  A  150-kw.  dynamo  was  supplying  273  amperes.  What 
was  the  voltage  of  the  dynamo  ? 

6.  How  many  horse  power  are  required  to  send  a  current  of 
65  amperes  through  10  mi.  of  No.  6  B.  &  S.  gauge  copper  wire  ? 

7.  A  current  of  15  amperes  flows  through  100  ft.  of  iron 
wire  whose  resistance  is  ^  ohm  per  foot.  How  many  watts  are 
lost  in  the  wire  ? 

8.  With  a  current  of  50  amperes  450  watts  are  absorbed  in 
the  conductor.    Find  the  drop  in  voltage  in  the  conductor. 

9.  A  voltmeter  has  a  resistance  of  17,000  ohms.  If  placed 
in  a  circuit  of  110  volts,  how  much  power  is  required  to 
operate  it  ? 

10.  A  200-volt  lamp  takes  ^  ampere.  How  many  watts  are  re- 
quired for  30  such  lamps  ?  How  many  horse  power  are  required 
to  drive  the  dynamo  if  it  has  an  efficiency  of  90  per  cent  ? 


ELECTRICITY  229 

Solution.       200  x  ^  x  30  =  2000  watts. 

— - —  =  2.68  h.  p.  100  per  cent  efficiency. 

2  68 

— —  =  2.98  h.  p.  90  per  cent  efficiency. 

11.  In  a  room  there  are  thirty  16  candle  power  incandescent 
lamps,  each  taking  .52  amj)ere  at  110  volts ;  and  3  arc  lights, 
each  taking  6.8  amperes  at  50  volts.  How  many  watts  and  how 
many  horse  power  are  required  to  operate  these  lights  ? 

12.  How  many  incandescent  lamps,  each  having  a  resistance 
of  220  ohms  and  requiring  a  current  of  .5  ampere  can  be  run 
by  a  10-kw.  generator  ? 

13.  A  25  h.  p.  dynamo  is  running  at  550  volts.  How  many 
amperes  in  the  current  ?  How  many  16  candle  power  incandes- 
cent lamps  can  be  placed  on  the  circuit  if  each  lamp  takes 
55  watts  and  there  is  a  loss  of  10  per  cent  on  the  line  ? 

.  14.  In  an  electric-lighting  circuit  there  are  60  arc  lights,  each 
taking  50  volts,  and  15  mi,  of  wire  having  a  resistance  of 
2.1  ohms  per  mile.  If  the  current  is  9.6  amperes,  how  many 
watts  are  required  to  run  the  lights  ? 

15.  Find  the  energy  in  foot  pounds  expended  per  candle 
power  in  a  16  candle  power  incandescent  lamp  in  1  hr.,  if  it 
takes  ^  ampere  at  110  volts. 

16.  If  a  500  candle  power  arc  light  requires  50  volts  with 
9.6  amperes,  how  many  foot  pounds  per  candle  power  are  ex- 
pended in  1  hr.  ?  How  does  this  compare  with  the  result  in 
Problem  15  ? 

17.  A  10-kw.  dynamo  has  an  efficiency  of  88  per  cent.  How 
many  horse  power  are  required  to  drive  it  ? 

18.  The  efficiency  of  a  dynamo  is  85  per  cent.  How  many 
horse  power  are  required  to  drive  it  when  there  are  200 
16  candle  power  lamps  on  the  circuit,  each  lamp  taking  ^ 
ampere  at  110  volts  ? 


230  APPLIED  MATHEMATICS 

19.  How  many  amperes  at  120  volts  must  be  furnished  a 
hoisting  motor  which  is  to  lift  900  lb.  70  ft.  per  minute,  if  it 
has  an  efficiency  of  70  per  cent  ? 

20.  A  motor  operates  a  pump  which  in  1  hr.  lifts  20,000  gal. 
of  water  (1  gal.  =  8i-  lb.)  400  ft.  If  the  combined  efficiency  of 
the  pumping  system  is  72  per  cent,  what  current  will  the  motor 
require  at  550  volts  ? 

21.  An  electric  street  car  with  its  load  weighs  8  T. ;  on  a 
level  track  the  pull  required  is  20  lb.  per  ton.  How  much  power 
is  necessary  at  the  axle  to  move  the  car  10  mi.  per  hour  ?  If 
the  motor  and  gearing  have  an  efficiency  of  75  per  cent,  how 
many  amperes  are  required  on  a  550-volt  circuit  ? 

22.  To  perform  a  certain  amount  of  work  30  kw.  hr,  are 
required.  If  the  dynamo  gives  a  current  of  125  amperes  at 
220  volts,  how  long  must  it  be  used  to  perform  this  work  ? 

o  125  X  220      „_  _ . 

Solution.  — =  27.5  kw. 

1000 

-^  =  1.09  hr. 
27.5 

23.  A  5-kw.  motor  is  used  to  operate  a  printing  press  8  hr. 
What  will  be  the  cost  of  the  power  at  12  cents  per  kilowatt 
hour  ? 

24.  What  is  the  cost  of  running  a  motor  which  requires 
15  amperes  at  110  volts,  at  12  cents  per  kilowatt  hour  ? 

25.  An  incandescent  lamp  takes  .6  ampere  at  110  volts.  If 
power  costs  15  cents  per  kilowatt  hour,  what  is  the  cost  of 
operating  the  lamp  12  hr.  ? 

26.  An  inclosed  arc  lamp  takes  80  volts  on  a  current  of  6.6 
amperes.  How  much  does  it  cost  to  operate  the  lamp  12  hr. 
at  15  cents  per  kilowatt  hour  ? 

27.  How  many  watts  per  candle  power  are  required  in  each 
of  the  following  lamps  ?  If  power  costs  10  cents  per  kilowatt 
hour,  how  much  would  it  cost  per  hour  to  keep  each  lamp  at 


ELECTRICITY 


231 


full  candle  power?  Construct  a  curve  to  show  the  relation, 
between  the  candle  power  of  each  lamp  and  the  cost  per 
candle  power. 


Volts 

Candle  power 

Amperes 

Ohms 

110 

10 

.32 

344 

110 

16 

.51 

216 

110 

20 

.64 

172 

110 

24 

.76 

145 

110 

32 

1.02 

108 

28.  From  the  equation  ]'  x  A  =  W  construct  a  series  of 
curves  on  the  same  axes  for  W  equal  to  1,  2,  3,  4,  5  kw.  Know- 
ing the  voltage  and  current  in  a  circuit,  by  means  of  these 
curves  the  approximate  power  can  be  determined  readily. 


Heat  generated  by  a  Current 

126.  Heat  loss  in  a  conductor.  We  have  seen  that  it  takes 
pressure  (voltage)  to  drive  a  current  through  a  conductor,  and 
we  have  computed  this  fall  of  potential.  Thus  if  a  current  of 
10  amperes  flows  through  a  resistance  of  2  ohms,  the  amount 
of  voltage  required  to  send  the  current  isF  =  ^ii=10  x  2  = 
20  volts.  We  have  also  computed  the  loss  of  power.  Thus  the 
number  of  watts  lost  is  F  x  ^  =  200  watts.  This  power  or 
energy  which  is  lost  in  the  conductor  is  changed  into  heat. 
We  may  say  in  the  above  problem  that  the  heat  loss  is  200 
watts  per  minute. 

Hence  to  find  the  heat  loss  in  a  conductor  we  simply  find  the 
watts  lost,  and,  if  desired,  change  the  watts  into  calories  or 
British  thermal  units. 

1  watt  minute  =  44.25  ft.  lb.  per  minute. 
1  B.  t.  u.  =  778  ft.  lb. 
Hence  1  watt  minute  =  .057  B.  t.  u.  per  minute. 


232  APPLIED  MATHEMATICS 

PROBLEMS 

1.  Find  the  heat  loss  due  to  a  current  of  60  amperes  through 
a  resistance  of  10  ohms. 

Solution.    W  =  A^  ■  R  =  60^  x  10  =  36,000  watts  in  1  min. 

2.  A  conductor  having  a  resistance  of  5  ohms  carries  a  cur- 
rent of  18  amperes.    How  much  heat  is  developed  in  1  hr.  ? 

3.  How  much  heat  is  developed  in  a  wire  of  15.2  ohms  re- 
sistance by  a  current  of  8  amperes  in  15  min.,  («)  in  watts  ? 
(b)  in  calories  ?  (c)  in  Bi'itish  thermal  units  ? 

4.  A  current  of  36  amperes  is  sent  over  a  line  of  2  ohms 
resistance.  What  is  the  drop  in  voltage  ?  What  is  the  heat 
loss  per  hour  (a)  in  watts  ?  (b)  in  British  thermal  units  ? 

5.  A  current  of  12  amperes  flows  through  a  resistance  of 
3.2  ohms  for  15  min.,  and  another  current  of  8  amperes  flows 
through  a  conductor  of  2.5  ohms  resistance.  How  long  must 
the  second  current  flow  in  order  that  the  amount  of  heat  gen- 
erated may  be  the  same  as  in  the  first  case  ? 

6.  Construct  a  curve  to  show  the  heat  loss  in  a  conductor 
as  the  resistance  changes  from  1  to  10  ohms  while  the  current 
remains  constantly  5  amperes. 

7.  Construct  a  curve  to  show  the  heat  loss  in  a  conductor 
of  1  ohm  resistance  as  the  current  varies  from  10  to  20  amperes. 

8.  In  a  conductor  of  10  ohms  resistance  the  voltage  increases 
from  10  to  1000  volts.    Construct  a  curve  to  show  the  heat  loss. 

9.  A  Leclanche  cell  used  to  ring  a  doorbell  has  an  electro- 
motive force  of  1.6  volts  and  the  current  is  .75  ampere.  If  the 
wire  has  a  resistance  of  .4  ohm,  what  per  cent  of  the  power 
is  the  heat  loss  in  the  line  ? 

Solution.  1.6  x  .75  =  1.2  watts,  total  power. 

A^R  =  .752  X  .4  =  .23  watts,  heat  loss  in  line. 

—  =  19  per  cent. 
1.2  ^ 


ELECTRICITY  233 

10.  The  dynamo  of  an  arc-light  system  furnishes  a  current 
of  9.6  amperes  at  3000  volts.  The  circuit  is  made  up  of  16  mi. 
of  No.  6  B.  &  S.  gauge  copper  wire.  What  per  cent  of  the  power 
is  the  heat  loss  in  the  line  ? 

11.  A  6-in.-plate  stove  requires  5.5  amperes  at  110  volts. 
What  is  the  cost  of  running  it  30  min.  if  the  current  costs 
6  cents  per  kilowatt  hour  ? 

12.  Find  the  cost  of  heating  a  6^-lb.  flatiron  for  3  hr.,  if  it 
takes  4  amperes  at  110  volts,  at  6  cents  per  kilowatt  hour, 

13.  An  electric  radiator  takes  13.6  amperes  at  110  volts. 
Find  the  cost  for  8  hr.  at  6  cents  per  kilowatt  hour. 

14.  In  an  electric  heater  there  is  a  coil  of  iron  wire  224  ft, 
in  length  having  a  resistance  of  ^  ohm  per  foot.  .  If  it  is  con- 
nected to  a  110-volt  circuit,  how  much  heat  is  generated  ? 

15.  It  is  desired  to  make  an  electric  soldering  iron  to  be 
heated  by  a  coil  of  No.  27  German  silver  wire  of  resistance 
1.25  ohms  per  foot.  How  many  feet  will  be  required  to  give 
200  watts  on  a  500-volt  circuit  ? 

Wiring  for  Light  and  Power 

127.  The  mil.  In  electrical  calculations  involving  the  diam- 
eter of  wire,  the  mil  is  usually  taken  as  the  unit  of  length. 

1  mil  =  .001  in. 

A  circular  mil  is  a  circle  whose  diameter  is  1  mil. 

A  circular  mil  =  m-r^  =  tt  x  .5^  =  .7854  sq.  mils. 
1  circular  mil  =  .7854  sq.  mils. 

1  sq.  mil  =  1.273  circular  mils. 

Circles  are  to  each  other  as  the  squares  of  their  diameters. 
Hence  to  find  the  area  of  a  circle  in  circular  mils,  square  its 
diameter  expressed  in  mils. 

A  mil  foot  of  wire  is  1  ft.  long  and  1  mil  in  diameter.  In 
practice  the  resistance  of  1  mil-ft.  of  copper  wire  is  usually 
taken  as  10.7  ohms. 


234  APPLIED  MATHEMATICS 

PROBLEMS 

1.  The  diameter  of  a  wire  is  i  in.  Find  (a)  its  diameter  in 
mils ;  (b)  its  cross  section  in  circular  mils. 

2.  How  many  circular  mils  in  the  cross  section  of  a  wire  of 
diameter  (a)  J  in.  ?  (b)  .125  in.  ?  (c)  .06  in.  ? 

3.  Find  the  diameter  and  area  in  square  mils  of  a  wire 
whose  cross  section  is  (a)  10,381  circular  mils ;  (b)  26,250 
circular  mils ;  (c)  105,590  circular  mils. 

4.  A  copper  bar  is  1  in.  by  J  in.  Find  the  area  of  a  cross 
section  in  square  mils  and  in  circular  mils. 

5.  Find  the  resistance  of  1000  ft.  of  copper  wire  40  mils  in 
diameter. 

Solution.    The  cross  section  =  40^  =  1600  circular  mils. 

Resistance  of  1  mil  foot  =  10.7  ohms. 

Resistance  of  1  ft.  of  wire  of  40  mils  diameter 

10  7 
=  i±lii-  =  .00669  ohm. 
1600 

Resistance  of  1000  ft.  of  wire  of  40  mils  diameter 

=  .00669  X  1000  =  6.69  ohms. 

6.  Find  the  resistance  of  1000  ft.  of  copper  wire  that  has  a 
diameter  of  (a)  460  mils ;  (b)  289.3  mils ;  (c)  .1  in. ;  (d)  40.3 
mils  ;  (e)  20  mils. 

7.  A  current  of  75  amperes  is  sent  through  1  mi.  of  copper 
wire  229.4  mils  in  diameter.    Find  the  drop  in  voltage. 

Solution.    Cross  section  =  229.4'^  =  52,620  circular  mils. 

10  7 
Resistance  of  1  ft.  =  =  .000203  ohm. 

52,620 

Resistance  of  1  mile  =  .000203  x  5280 

=  1.07  ohms. 

V=A-R  =  75x  1.07  =  80.3  volts. 

8.  What  is  the  drop  in  voltage  in  a  circuit  of  5  mi.  of  copper 
wire  162  mils  in  diameter  if  the  current  is  40  amperes  ? 


ELECTRICITY  235 

9.  With  a  current  of  210  amperes  what  will  be  the  drop  in 
voltage  in  2500  ft.  of  copper  wire  460  mils  in  diameter  ? 

10.  How  many  circular  mils  are  required  in  a  power  line 
500  ft.  long  with  a  current  of  150  amperes,  if  a  drop  of  12  volts 
is  allowed  ? 

Solution.    Let  n  =  number  of  circular  mils  required. 

10.7  X  500  =  5350  ohms  per  circular  mil  for  500  ft. 

5350 

=  number  of  ohms  resistance  of  n  circular  mils 

"  for  500  ft. 

5350  X  150  ,         ,      1.    .      , 

=  number  of  volts  in  drop. 

n 

5350  X  150  _ 

•'  *  n 

n  —  66,880  circular  mils. 

10.7 

Check.         =  .00016  ohm,  resistance  of  1  ft.  of  the  wire. 

66,880 

.00016  X  500  =  .08  ohm,  resistance  of  500  ft. 

V=AR  =  \bO  X  .08  =  12. 

From  the  above  solution  we  may  obtain  the  following  formula, 

which  is  in  general  use  for  finding  the  size  of  conductor  required 

to  carry  a  given  load. 

„      .      ,         .,        21.4  X  distance  one  way  in  feet  x  amperes 

JSo.  circular  mils  = r; — r—, 

volts  lost 

11.  A  motor  is  300  ft.  from  the  dynamo.  How  many  circular 
mils  are  required  for  a  current  of  90  amperes,  if  a  drop  of 
6  volts  is  allowed  ? 

12.  Find  the  number  of  circular  mils  required  to  deliver 
10  kw.  to  a  motor  at  a  distance  of  200  ft.,  with  100  volts  pressure 
at  the  motor,  if  a  drop  of  5  volts  is  allowed  in  the  line. 

13.  Find  the  number  of  circular  mils  required  to  transmit 
25  kw.,  with  a  20  per  cent  drop  in  the  voltage,  a  distance  of 
10  mi.,  if  the  voltage  at  the  load  is  to  be  (a)  100  volts  ;  (Ji)  500 
volts ;  (c)  1000  volts. 

Suggestion.  -—  =  125  volts  at  the  dynamo. 

.80 


236 


APPLIED  MATHEMATICS 


14.  It  is  required  to  deliver  120  h.  p.  to  a  motor  2  mi.  away, 
from  a  dynamo  which  has  a  voltage  of  550  volts.  If  the  line 
loss  is  to  be  not  more  than  16  per  cent,  find  the  cross  section 
of  the  wire  in  circular  mils  and  the  number  of  pounds  of 
copper  required. 

Dynamos  and  Motors 

128.  Construction.  When  a  closed  wire  is  rotated  between 
the  poles  of  a  magnet  so  as  to  cut  the  lines  of  force,  a  current 
ilows  in  the  wire.  The  dynamo  is 
constructed  on  this  principle.  The 
avTnature  is  the  part  of  the  ma- 
chine in  which  the  current  is  gen- 
erated, and  in  most  machines  the 
armature  revolves.  The  field  is 
the  space  between  the  poles  of  the 
magnets   in  which   the    armature 

revolves.  The  magnets  are  pieces  of  soft  iron,  which  are  mag- 
netized by  a  current  from  the  machine  itself  or  from  a  sepa- 
rate dynamo.  This  current  flows  in  coils  which  are  placed 
around  the  magnets.  In  Fig.  96  the  armature  is  represented 
by  a  single  wire  revolving  in  the  field. 

129.  The  field  coils  connected  in  three  ways.  Direct  current 
dynamos  generally  excite  their  own  fields ;  and  there  are  three 
ways  of  connecting  the  field-magnet  coils. 

1.  Series-wound  dynamos.  The  field-magnet 
coils  are  connected  to  the  armature  so  that 
the  whole  current  generated  passes  through 
them. 

2.  Shunt-ivound  dynamos.  The  field-mag- 
net coils  are  connected  in  multiple  with  the 
terminals  of  the  armature ;  hence  only  a 
part  of  the  current  goes  through  them. 
These  coils  consist  of  many  turns  of  comparatively  fine  wire. 


Fig.  97 


Fig. 


ELECTRICITY  237 

3.   Compound-wound  dynamos.    The  field  magnets  are  wound 
with  two  sets  of  coils,  one  in  series  and 
one  in  multiple  with  the  armature. 

Motors  are  also  wound  in  these  three 

ways. 

Fig.  99 

130.  Electrical  efficiency  of  dynamos 
and  motors.  Since  it  requires  pressure  (voltage)  to  drive  a 
current  through  the  armature  and  field  coils,  there  is  a  loss 
of  power  in  a  dynamo  and  in  a  motor.  This  loss  is  sometimes 
called  the  copper  loss.  Electrical  efficiency  takes  into  account 
only  the  copper  loss. 

_,        .    ,    «,  •  r.      -1  Power  given  out 

Electrical  efliciency  oi  a  dynamo  =  =r — — ^ — —  • 

•  Fower  generated 

^,        .    ,     „  .  „  ,  Power  left  for  useful  work 

Electrical  emciency  ot  a  motor     =  -=- r— r- 

Power  supplied  to  motor 


PROBLEMS 

1.  The  output  of  a  series-wound  dynamo  is  5  kw.  at  a  voltage 
of  110  volts.  The  resistance  of  the  armature  is  .06  ohm  and  of 
the  field  coil  .072  ohm.  Find  (a)  the  copper  loss ;  (b)  the  elec- 
trical efficiency ;  (c)  the  total  electromotive  force  generated. 

Solution.  i:  ^^-^ 

W      5000      ,,,  ^ ^OOOy^^rr^ 

45.5  amperes. 


V        110 
.06  +  .072  =  .132  ohm,  total  resistance 

(a)  Am  = 
45.5^  X  .132  =  273  watts,  copper  loss.  Fig.  lOo' 

5000  +  273  =  5273,  total  power  generated. 

(b)  llyt  =  95  per  cent,  electrical  efficiency. 

(c)  V=AR  = 

45.5  X  .132  =  6  volts,  loss  in  armature  and  field  coils. 
110  +  6  =  116,  total  electromotive  force  generated. 

2.  A  series-wound  motor  has  a  resistance  of  .68  ohm.    When 
supplied  with  15  amperes    at   a    voltage    of  105  volts,    find 


238 


APPLIED  MATHEMATICS 


(a)  the  copper  loss ;   (b)  the  electrical  efficiency ;   (c)  the  volts 
lost  in  the  motor. 

Suggestion.  Find  the  copper  loss  as  in  Problem  1  and  subtract  it 
from  the  number  of  watts  supplied  to  the  motor. 

Electrical  efficiency  =  ^|ff •    Drop  in  voltage  =  15  x  .68. 

3.  A  shunt-wound  dynamo  furnishes  5  kw.  at  a  voltage  of 
110  volts.  The  shunt  resistance  is  45  ohms  and  the  armature 
resistance  is  .06  ohm.  Find  (a)  the  copper  loss ;  (b)  the  electrical 
efficiency. 

Solution. 


W 


A  =  —  =  45..5  amperes. 


3000tr/l7-W 


In  the  shunt, 

-1= 


/JOyOLTtS 


2.44  amperes. 


110 
45 

45.5  +  2.44  =  47.9  amperes. 
IT  =  F^  =  110  X  2.44  =  268  watts,  loss  in  shunt. 
W  =  A^Pi.  =  47.92  X  .06  =  137  watts,  loss  in  armature. 
(a)  405  watts,  total  loss. 

5000  +  405  =  5405  total  watts. 
(6)  f  f  ^f  —  93  per  cent,  electrical  efficiency. 

4.  The  armature  of  a  shunt  motor  has  a  resistance  of  .02  ohm, 
and  the  shunt  a  resistance  of  62  ohms.  If  the  input  is  5  h.  p. 
at  124  volts,  find  (a)  the  copper  loss  ;  (b)  the  electrical  efficiency. 


Solution. 
A 


0  X  746  =  3730  watts. 
W     3730 


124 


=  30.1  amperes. 


V      124 

In  shunt,   A  =  —  = =  2  amperes. 

R       62  ^ 


S/rrfi. 


Fig.  102 


30.1  -2  =  28.1  amperes. 

W  =  VA  =  124  X  2  =  248  watts,  loss  in  shunt. 

V  =  A'^R  =  28.12  X  .02  =  _16  watts,  loss  in  armature. 

(a)  264  watts,  total  loss. 

3780  -  264  =  3466  watts  for  useful  work. 

(b)  f  tf  f  ~  ^^  P^^  cent,  electrical  efficiency. 

Note  that  the  current  in  the  armature  of  a  shunt  motor  equals 
the  total  current  less  the  current  in  the  field  coils. 


ELECTRICITY  289 

5.  A  50-k-w.,  125-volt,  compound-womid  dynamo  has  a  shunt 
resistance  of  62.5  ohms,  a  series-coil  resistance  of  .001  ohm,  and 
an  armature  resistance  of  .002  ohm.  soooOkV/iTTS 
Compute  the  copper  losses  and  the      E^^^S^^^-v^       ** 
electrical  efficiency.                                   ,^<  ^oi\  ^^ /zsyoiJiS 

Solution.         -\W-  =  400  amperes.       **  ^   ^^-^^  ^ 

In  shunt,      — -  =  — -—  =  2  amperes.  ^^^-  ^^^ 

R      62.5 

400  +  2  =  402  amperes,  total  current  generated. 

4022  X  .002  =  323  watts,  loss  in  armature. 

402^  X  .001  =  162  watts,  loss  in  series  coil. 

125  X  2  =  250  watts,  loss  in  shunt. 

735  watts,  total  loss. 

f  TT?T  ~  ^^'^  P^''  cent,  electrical  efficiency. 

Note  that  the  total  current  generated  by  a  shunt  dynamo  equals 

the  sum  of  the  currents  in  the  armature  and  in  the  field  coils. 

6.  A  compound  motor  is  supplied  with  50  amperes  of  current 
from  110-volt  mains.  If  the  armature  resistance  is  .09  ohm,  the 
series-coil  resistance  .078  ohm,  and  the  shunt-coil  resistance 
55  ohms,  find  ia)  the  copper  loss ;  (Ji)  the  electrical  efficiency. 

Solution.  50  x  110  =  5500  watts. 

—  =  -—-  =  2  amperes  m  shunt. 
it       5o 

50  —  2  =  48  amperes  in  armature. 

Find  loss  in  shunt,  armature,  and  series  coil  to  be  220,  207,  and 

180  watts  respectively,  and  the  electrical  efficiency  89  per  cent. 

7.  The  output  of  a  series  dynamo  is  20  amperes  at  1000  volts. 
The  resistance  of  the  armature  is  1.4  ohms  and  of  the  field  coil 
1.7  ohms.  Find  the  copper  loss,  the  electrical  efficiency,  and 
the  volts  lost  in  the  dynamo. 

8.  The  armature  of  a  shunt  motor  has  a  resistance  of  .3  ohm, 
and  the  shunt  a  resistance  of  120  ohms.  When  running  at  full 
load  on  a  110-volt  circuit  the  motor  takes  a  current  of  8  amperes. 
Find  the  copper  loss  and  the  electrical  efficiency. 


240 


APPLIED  MATHEMATICS 


Find  the  copper  losses  and  electrical  efficiency  of  the  follow- 
ing dynamo-electric  machines : 

Dyxamos 


No 

Type 

Resistance,  ohms 

Output 

Volts 

Armature 

Series  coil 

Sliunt  coil 

9 
10 
11 
12 
13 
14 
15 
16 

Series 

Compound 

Shunt 

Series 

Compound 

Shunt 

Compound 

Shunt 

2 
.003 
.29 
.15 
.04 
.006 
.023 
.0117 

2.5 
.002 

.12 
.03 

.012 

55 
57.5 

20 
12 
19.4 
52.7 

10  kw. 
60  h.  p. 
6.5  kw. 

10  kw. 
50  kw. 

1000 
110 
115 
110 

*  110 

111 
410 

50 

600 
220 
590 

Motors 


Resistance,  ohms 

No. 

Type 

Input 

Volts 

Amperes 

Armature 

Series  coil 

Sliunt  coil 

17 

Shunt 

.15 

48 

110 

10 

18 

Series 

.39 

.35 

Ikw. 

80 

19 

Shunt 

.14 

44 

110 

50 

20 

Shunt 

.018 

200 

30  kw. 

400 

21 

Series 

.112 

.113 

220 

100 

22 

Compound 

.14 

.02 

55 

5.5  kw. 

110 

131.  Commercial  or  net  efficiency.  The  commercial  effi- 
ciency of  a  dynamo  or  motor  takes  account  of  all  the  losses  in 
the  machine ;  it  is  equal  to  the  output  divided  by  the  input. 

^  •  7    ^  •  Output 

Input 


•     ELECTRICITY  241 

PROBLEMS 

1.  A  motor  is  supplied  with  a  current  of  20  amperes  at 
110  volts.  If  2.8  h.  p.  are  developed  at  the  pulley,  find  the 
commercial  efficiency  of  the  motor. 

Solution.  Input  =  110  x  20  watts. 

Output  =  746  X  2.8  watts. 

Commercial  efficiency  =  — ; -^ 

^       110  X  20 

=  95  per  cent. 
Check.  110  X  20  X  .95  =  2090  watts  =  2.8  h.  p. 

2.  A  motor  generator  takes  a  current  of  14  amperes  at  220 
volts  and  supplies  a  cui-rent  of  112  amperes  at  25  volts.  Find 
its  efficiency. 

3.  A  220-volt  electric  hoist  is  raising  coal  at  the  rate  of  1  T. 
270  ft.  per  minute.  If  the  current  is  90  amperes,  what  is  the 
efficiency  of  the  hoist  ? 

4.  A  3-kw.  motor  is  used  to  operate  a  lathe.  Find  its  effi- 
ciency if  it  takes  30  amperes  at  110  volts. 

5.  The  output  of  a  generator  is  50  kw.  If  it  requires  76  h.  p. 
to  drive  it,  what  is  its  efficiency  ? 

6.  A  550-volt  generator  supplies  a  current  of  300  amperes. 
If  the  generator  has  an  efficiency  of  85  per  cent,  how  many 
horse  power  are  required  to  drive  it  ? 

7.  It  takes  25  h.  p.  to  operate  a  dynamo  which  supplies  power 
for  40  arc  lights  in  series  at  7  amperes.  The  resistance  of  each 
lamp  is  8  ohms  and  the  line  resistance  is  25  ohms.  Find  the 
efficiency  of  the  dynamo. 

8.  A  lighting  circuit  consists  of  1200  ft.  of  No.  6  B.  &  S, 
gauge  copper  wire  and  eighty  16  candle  power  incandescent 
lamps  in  multiple,  each  having  a  resistance  of  220  ohms.  If 
the  voltage  is  110  at  the  lamps  and  7.5  h.  p.  is  supplied  to 
the  generator,  find  its  efficiency. 


242 


APPLIED  MATHEMATICS 


9,  In  testing  a  motor  the  following  results  were  obtained. 
Find  the  efficiency  given  by  each  test. 


No. 

Volts 

Amperes 

Brake  horse  power 

1 

224 

96.6 

24.6 

2 

221 

101 

25.7 

3 

222 

103 

27.2 

4 

230 

109 

29.1 

6 

227 

123 

32.6 

10.  The  following  data  were  obtained  in  a  test  of  a  motor 
generator. 

Construct  a  curve  showing  the  relation  between  output  and 
efficiency. 


Input 

Volts 
Amperes 

226 
6.9 

225 

7.7 

229 
9.6 

228 
11.7 

228 
13.7 

228 
16.9 

Output 

Volts 
Amperes 

21 
0 

20.8 
20 

21 
40 

20.6 
60 

20.2 
80 

20 
100 

CHAPTER  XVIII 

LOGARITHMIC  PAPER 

132.  Description  of  logarithmic  paper.  In  many  engineering 
problems  where  it  is  necessary  to  compute  a  set  of  values  from 
a  formula,  it  is  found  that  the  required  values  can  be  secured 
quickly  and  easily  by  using  paper  ruled  on  the  logarithmic 
scale.  This  paper  is  used  both  as  a  "  ready  reckoner,"  to  read 
off  tables  of  values  and  to  find  the  law  connecting  the  two 
variables  in  the  problem.  The  advantage  of  logarithmic  paper 
lies  in  the  fact  that  many  formulas  which  are  represented  by 
curves  on  squared  paper  are  represented  by  ^straight  lines  on 
logarithmic  paper.  Hence  while  many  pairs  of  values  must  be 
worked  out  to  construct  a  curve  on  the  former,  only  two  or 
three  pairs  are  required  for  the  latter. 

Fig.  104  shows  the  way  in  which  logarithmic  paper  is  ruled. 
The  cc-axis  and  the  ?/-axis  are  laid  off  in  divisions  exactly  like 
those  of  the  slide  rule.  That  is,  OX  and  OF  are  each  divided 
into  1000  equal  parts ;  2  is  placed  at  the  301st  division 
(log  2  =  0.301)  ;  3  is  placed  at  the  477th  division  (log  3  = 
0.477) ;  4  is  placed  at  the  602d  division  (log  4  =  0.602), 
and  so  on. 

Exercise.  Construct  a  graph  to  read  off  the  area  of  a  circle  of 
any  given  radius. 

In  order  to  learn  the  properties  of  logarithmic  paper  we  will 
construct  the  graph  by  locating  points.  Later  it  will  be  shown 
that  the  whole  graph  can  be  constructed  easily  by  locating  only 
one  point. 

243 


244  APPLIED  MATHEMATICS 

The  formula  for  the  area,  a  =  irr^,  gives  the  following  table 


Radius 
Area  . 


1 
3.14 


1.2 
4.52 


1.5 
7.07 


2 
12.6 


3 
28.3 


4 

50.3 


5 

78.5 


6 
113 


154 


201 


10 
314 


7 8    |9    [0^ 


Fig.  104 


Locating  the  points  as  shown  in  Fig.  104,  we  see  that  the 
points  lie  on  the  straight  lines  AB,  CD,  and  EF.  Hence 
AB  —  CD  —  EF  is  the  graph  required.  From  it  we  see  that 
when  the  radius  is  2.5  the  area  is  19.6 ;  when  the  area  is  38.5, 
the  radius  is  3.5,  and  so  on. 


LOGARITHMIC  PAPER 


245 


133.  Properties  of  logarithmic  paper.    Some  properties  of 

the  paper  may  now  be  noted.    The  equation  a  =  irr^  is  in  the 

form  ij  =  mx"*.    AB,  CD,  and  EF  are  parallel  to  one  another. 

FX 
BD  =  CE  =  ^  YZ.  FX  =  2 EX;  hence  TT^  =  2,  the  exponent  of  r. 

The  graph  can  be  drawn  mechanically  as  follows :  Find  P, 
the  mid-point  of  YZ.  Tack  the  sheet  of  paper  on  a  drawing 
board  so  that  the  T-square,  in  position,  lies  on  O  and  P.  Set 
the  T-square  on  A  (making  OA  =  3.14)  and  draw  AJ3.  Set  the 
T-square  at  C  on  OX  directly  below  B  and  draw  CD.  Similarly, 
draw  EF.  Check;  F  should  be  directly  opposite  A,  that. is, 
FX  =  3.14. 

It  will  be  found  that  these  are  general  properties  of  logarith- 
mic paper,  which  may  be  used  to  construct  graphs  for  formulas 
of  the  form  y  =  mx'' ;  that  is,  a  formula  in  which  y  equals  an 
expression  consisting  of  only  one  term  in  which  the  variable 
is  raised  to  any  power  (n,  being  positive,  negative,  or  fractional) 
and  multiplied  by  any  number.  This  form  alone  will  be  con- 
sidered in  the  following  discussion,  and  some  of  the  properties 
of  the  paper  which  lead  to  simple  and  accurate  constructions 
will  be  considered. 

I.   Equations  of  the  Form  y  =  mx 


EXERCISES 
1.  Construct  the  graph  of  y  =  x. 


X 

1 

2 

3 

4 

5 

y 

1 

2 

3 

4 

5 

Locating  the  points  from  the  table,  we  see  that  they  lie  on  the 
straight  line  OZ  (Fig.  105).    Hence  OZ  is  the  graph  of  y  =  x. 

2.  Construct  on  the  same  sheet  of  paper  the  graph  of  (1)  y  = 
2x;(2)y  =  3x;{3)y^4.x. 


246 


APPLIED  MATHEMATICS 


It  is  seen  that  all  these  lines  are  parallel.  When  we  plot 
2/  =  cc  (1)  and  y  =  2x  (2),  we  are  really  plotting  the  logarith- 
mic equations  log  y  =  log  x  (!')  and  log  y  =  log  2x,  or  log  y  = 
logic  +  log  2  (2').    Comparing  (!')  and  (2'),  we  see  that  they 


V 

>J 

F 

P 

z: 

ID                                                • 

/ 

/ 

/ 

9                              / 

/ 

/ 

/ 

8                                               y/^ 

/ 

/ 

/ 

-•                   f 

/ 

/ 

/ 

M 

/ 

/ 

Q 

s          y 

/ 

/ 

/ 

/ 

E 
A 

0 

A             /                            / 

/^ 

■/ 

/ 

/ 

3        y 

r 

/ 

/ 

/ 

/ 

/ 

0 

z                                 / 
X                                          P 

/ 

y 

/ 

c 

f 

/ 

/ 

/ 

/ 

/ 

2 

a            ■" 

\ 

5        « 

o 

1     i 

3     1 

o 

Fig.  105 


differ  only  by  the  constant  term  log  2  on  the  right  side ;  that 
is,  every  point  of  the  graph  of  (2)  is  2  above  the  correspond- 
ing point  of  the  graph  of  (1).  Note  that  the  graph  of  each  of 
these  equations,  except  y  =  a;,  is  made  up  of  two  lines  ;  and  all 
the  lines  are  parallel  to  OZ.  Hence  to  graph  any  equation  of 
the  form  y  =  ma;,  for  example,  y  =  hx,  proceed  as  follows.  From 


LOGARITHMIC  PAPER 


247 


5  on  OF  draw  MN  parallel  to  OZ.    Take  OP  =  YN  and  draw 
PQ  from  P  to  5  on  XZ.   MN—PQ  is  the  required  graph. 

The  slope  of  a  graph.  We  shall  find  that  each  graph  we  are 
to  consider  (except  y  =  x  and  y  =  x~'^)  consists  of  two  or  more 
parallel  lines,  and  that  one  line  in  each  graph  cuts  OX  and  XZ 
or  OX  and  OY.    Thus  in  the  graph  ot  y  =  5x,  PQ  cuts  OX  and 

XQ 
XZ.    We  will  call  — —  the  slope  of  the  graph  ;  that  is,  the  tan- 
XP 

gent  of  the  angle  which  the  line  makes  with  OX,  always  taking 

the  angle  on  the  right-hand  side  of  the  line. 

II.   Equations  of  the  Form  y  ^  ma;" 
A.    When  n  is  a  positive  whole  number. 


EXERCISES 
1.  Construct  the  graph  oi  y  =  x^. 


X 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

V 

1 

4 

9 

16 

25 

36 

49 

64 

81, 

100 

Locating  these  points,  we  get  the  graph  OA  —BZ  (Fig.  106). 
Note  that  A  and  B  are  the  mid-points  of  YZ  and  OX  respectively. 

2.  Construct  the  graph  of  y  —  a;^ 

Locating  points,  we  get  OD  —  FG  —  HZ  (Fig.  106).  Note  that 
i)and  G,  and  Fand  H  divide  YZ  and  OX  respectively  into  three 
equal  parts. 

3.  Construct  the  graphs  oty  =  x*  and  y  =  cc®  without  locating 
points. 

Roots  of  numbers.  From  the  graphs  ot  y  =  x^,  y  =  x',  y  =  a;*, 
and  so  on  we  can  read  ofE  roots  of  numbers.  Thus  in  the  graph 
oi  y  =  x^,  OA  gives  the  square  roots  of  numbers  from  1  to 
9,  100  to  999,  10,000  to  99,999, . . . ;  that  is,  of  numbers  con- 
taining 1,  3,  5  •  •  •  figures.  BZ  gives  the  square  root  of  numbers 
containing  2,  4,  6  •  •  •  figures.   To  find  the  square  root  of  2,  read 


248 


APPLIED  MATHEMATICS 


from  2  on  OY  to  OA,  1.41 ;  for  the  square  root  of  20  read  from 
2  on  OF  to  BZ,  4.47.  Similarly,  y  =  x^  gives  cube  roots;  OD 
gives  the  cube  root  of  numbers  containing  1,  4,  7  •  •  •  figures,  FG 


Y 

D 

A 

G 

lo 

/ 

/ 

/ 

9 

/           / 

/ 

a                                   y 

/ 

/ 

/ 

7                                          / 

/    , 

1 

i 

\ 

6                                    / 

/ 

/ 

1 

5                           / 

/ 

/ 

\ 

/   V 

■>  / 
If 

A 

1 
1 

1 

3                     /          / 

J\ 

i 

■J 

Z            /     / 

/\ 

/ 
1 

f 

X 

1                                     : 

l6          -^ 

k        w  ■! 

i           ( 

'        i 

i      S 

)     1 

3 

Fig.  106 


of  numbers  containing  2,  5,  8  •  •  •  figures,  and  HZ  of  numbers 
containing  3,  6,  9  •  •  •  figures. 

4.  Construct  the  graph  oi  y  =  2x^. 


X 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

V 

2 

8 

18 

32 

50 

72 

98 

128 

162 

200 

LOGARITHMIC  PAPER 


249 


Note  that  each  y  is  twice  as  great  as  the  corresponding  y  in 
y  =  x^.  On  locating  the  points  and  drawing  the  lines  of  the 
graph  it  will  be  seen  that  the  lines  are  parallel  to  the  lines  of 
2/  =  ic^  and  2  units  above  them.  Hence  the  graph  oi  y  =  2x^ 
may  be  constructed  mechanically  as  follows  :  Tack  the  sheet  of 
paper  on  a  drawing  board  so  that  the  edge  of  the  T-square,  in 
position,  lies  on  O  and  the  mid-point  of  YZ.  Move  the  T-square 
up  to  2  on  OF  and  draw  a  line  from  2  to  YZ.  Move  the 
T-square  to  a  point  on  OX  directly  below  the  point  already 
determined  on  YZ  and  draw  a  line  to  YZ.  Continue  in  the 
same  manner  and  the  graph  will  end  at  2  on  XZ  if  accurately 
drawn.  This  method  holds  for  all  cases  where  a;"  has  a  coeffi- 
cient.  Note  that  the  exponent  of  x  is  the  slope  of  the  graph. 

5.  Construct  the  graph  of  (a)  y  =  1x^\  (b)  y  =  .5x^', 
(c)  y  =  1.68  x^ ;  (d)  y  =  .0625  x». 

B.    When  n  is  a  positive  fraction. 


EXERCISES 

1.  Construct  the  graph  oi  y  —  xl. 

X 

1 

4 

9 

16 

25 

36 

49 

64 

81 

100 

y 

1 

8 

27 

64 

125 

216 

343 

512 

729 

1000 

Locating  points  from  the  table,  we  get  the  graph  OA—BC  — 
DE  —  FZ  (Fig.  107).  A  study  of  the  graph  shows  that  it  could 
be  drawn  in  the  following  manner :  Divide  OX  and  YZ  each 
into  three  equal  parts  by  the  points  F,  B,  E,  and  A  ;  and  OY 
and  XZ  each  into  two  equal  parts  by  the  points  D  and  C.  Join 
0  to  A,  the  second  point  of  division  on  YZ.  This  gives  the 
correct  slope,  |.  Directly  below  A  is  B,  draw  BC ;  opposite  C  is 
D,  draw  DE ;  below  E  is  F,  djaw  FZ. 

A  similar  construction  holds  for  any  positive  fractional  value 
of  n.  Thus  for  y  =  x\,  divide  OX  and  YZ  each  into  3  (the 
numerator  of  the  exponent)  equal  parts,  and  OF  and  XZ  each 


250 


APPLIED  MATHEMATICS 


into  5  (the  denominator  of  the  exponent)  equal  parts,  and  join 
the  points  so  as  to  make  the  slope  f . 

If  X  has  a  coefficient,  for  example,  y  =  5  x\,  start  the  graph  at 
5  on  OF  and  draw  it  parallel  to  OA^  thus  making  the  slope  |. 


2.  Construct  the  graphs  of  (a)  y  z=zx\\  (h)  y  =  2  xi  \  {c)  y  =■ 
3 a;f ;  (d)  y  =  5 x^* ;  (e)  y  =  2.5 x^-^ ;  (f)l/  =  -06 x'-K 

3.  Construct  a  graph  to  show  the  distance  passed  over  by  a 
falling  body  in  1  to  10  sec. 

4.  Construct  graphs  to  find  (a)  the  surface,  (b)  the  volume 
of  spheres  of  radii  from  1  to  10  in. 


LOGARITHMIC  PAPER 
C.    When  n  is  negative. 

EXERCISES 


251 


\ 

1.  Construct  the 

graph  of  y 

D 

OTy  = 

X 

z 

\, 

8 

\>^ 

X 

N 

\\^ 

N 

^^1 

\\ 

^ 

1 

N 

5 

X 

\ 

F 

\ 

Sy                  s 

X. 

k^ 

N 

E 

3 

B 

N^ 

f^- 

\ 

^^ 

\ 

\ 

X 

\ 

\ 

\ 

A 

^^ 

^ 

N 

\ 

\ 

^ 

^ 

\ 

( 

3                                           '' 

-                  c 

i  c             ^ 

*      J 

>        « 

3 

T      € 

J        5 

3    '< 

jX 

Fig.  108 


X 

1 

2 

4 

8 

10 

y 

1 

.6 

.25 

.125 

.1 

Locating  the  points  from  the  table,  we  get  the  graph  FA' 
(Fig.  108).    The  graph  oi  y  =  mx~^  is  parallel  to  FA,  and  we 


252  APPLIED  MATHEMATICS 

begin  to  draw  it  from  m  on  OF.  Thus,  to  graph  y  —  Ax~^,  from 
4  on  OY  draw  a  line  parallel  to  YX  cutting  OX  at  a  point  A  ; 
from  B  on  YZ  directly  above  A  draw  a  line  parallel  to  FA' 
cutting  XZ  at  C. 

2.  Construct  the  graph  of  y  =  x-h 

Divide  OX  and  YZ  into  2  (numerator  of  the  exponent)  equal 
parts,  and  OY  and  XZ  into  3  (denominator  of  the  exponent) 
equal  parts.  Draw  lines  as  shown  in  Fig.  108,  and  we  get  the 
graph  YA-BC-DE-  FX. 

3.  Construct  the  graphs  of : 

(a)y  =  3x-\  (e)  y  =  2Sx-\ 

(b)  y  =  .5 a;- 2.  (/)  2/  =  125 x" ^■'. 

(c)  y  =  4:x-i.  {(/)  y=.006x-^\ 

(d)  2/  =  8 X- 1  (h)  y  =  2800 x-^^. 

PROBLEMS 

1.  If  in  a  gas  engine  the  gas  expands  without  gain  or  loss 
of  heat,  the  law  of  expansion  is  found  to  be  pv^-^^  =  3060.  Con- 
struct the  curve  to  show  the  pressure  as  the  volume  increases 
from  10  cu.  in.  to  26  cu.  in. 

Locate  only  one  point  (Fig.  109)  ;  when  v  =  10,  p  =  180. 
Mark  this  point  by  ^  on  OF.  The  exponent  of  v  is  —  |§§,  when 
the  equation  is  in  the  form  p  =  3060  v~'^-^^. 

Measure  OM  =  123  mm.  on  OY,  and  ON  =  100  mm.  on  OX. 
Tack  the  paper  on  a  drawing  board  so  that  the  T-square,  in 
position,  lies  on  M  and  N.  Move  the  T-square  to  A  and  draw 
AB.  Move  the  T-square  to  C  on  YZ  directly  above  B  and  draw 
CD.  AB—CD  is  the  graph;  from  this  graph  pressures  can  be 
read  off  for  volumes  from  10  cu.  in.  to  100  cu.  in. 

Given  that  steam  expands  without  gain  or  loss  of  heat; 
construct  graphs  on  logarithmic  paper  for  volumes  from  10  to 
100  cu.  in. : 

2.  j9vi"  =  3000.  4.  pv\i  =  3200.  6.  pv^  =  250. 

3.  pv^-^^  =  2840.  5.  ^wi-81  =  3420. 


LOGARITHMIC  PAPER 


253 


7.  The  diameter  d  of  wrought-iron  shafting  to  transmit  h 
horse  power  at  100  r.  p.  m.  is  given  hy  d  =  .85  h\.  Construct 
the  graph  and  make  a  table  for  horse  power  from  10  to  80. 


Y 

c 

y,v'» 

30€.0 

Z. 

9 

V 

8 

7 

A 

e 
5 

M 

3 

t 

(0 

? 
a 

\ 

\ 

y 

\ 

A 

\ 

\ 

Volume 

N 

\ 

\ 

\ 

\ 

o 

B          -' 

J                        C 

J                  * 

J.            i 

5          « 

9 

r     (. 

3      i 

^  >o 

Fig.  109 


8.  The  number  of  gallons  of  water  per  minute  flowing  over 
a  rectangular  weir  6  in.  wide  is  given  by  </  =  17.8  /ti,  where 
g  =  the  number  of  gallons  per  minute,  and  h  =  the  depth  in 
inches  from  the  level  of  free  water  to  the  sill  of  the  weir. 
Construct  the  graph  and  make  a  table  showing  the  nmnber  of 
gallons  per  minute  for  depths  1,  1.5,  2,  2.5,  •  •  • ,  6  in. 


254  APPLIED  MATHEMATICS 

9.  The  number  of  cubic  feet  of  water  per  minute  discharged 
over  a  V-notch,  or  triangular  weir,  is  given  by  Q  =  18.5  hhl, 
where  Q  =  the  number  of  cubic  feet  per  minute,  h  =  breadth  of 
notch  in  feet  at  the  free  surface,  and  h  =  depth  in  inches  from 
the  free  level  to  the  bottom  of  the  notch.  Construct  a  graph 
and  make  a  table  for  the  quantity  of  water  discharged  for 
depths  from  6  to  15  in.  when  Z»  =  1  ft. 

10.  The  diameter  of  a  copper  wire  which  will  be  fused  by  an 
electric  current  is  given  by  d  —  .00212  Ai,  where  d  =  the  diam- 
eter in  inches,  and  A  =  the  number  of  amperes.  Construct  a 
graph  and  make  a  table  of  diameters  of  wire  which  will  be 
fused  by  currents  of  10,  20,  30,  •  •  • ,  100  amperes. 

11.  The  weight  in  poimds  that  a  rectangular  steel  beam, 
supported  at  both  ends,  can  sustain  at  its  center  is  given  by 

hd^ 
w  —  890  —  >  where  tv  =  the  weight  in  pounds,  b  =  the  breadth 

of  beam  in  inches,  d  =  the  depth  of  beam  in  inches,  and  I  =  the 
length  of  beam  in  feet. 

Find  the  number  of  pounds  that  can  be  supported  at  the 
middle  of  a  steel  beam  4  in.  in  breadth  and  15  ft.  long  for 
depths  from  4  to  10  in. 

12.  In  accordance  with  the  building  laws  of  Chicago  the  safe 
load  in  tons,  uniformly  distributed,  for  yellow-pine  beams  is 

given  hj  w  =  '- — - —  >  where  w  =  load  in  tons,  b  =  breadth  of 

beam  in  inches,  d  =  depth  of  beam  in  inches,  and  I  =  length  of 
beam  in  feet  between  the  supports. 

Find  the  safe  load  for  yellow-pine  beams  25  ft.  long,  4  in. 
in  breadth,  and  depths  from  8  to  18  in. 

13.  The  number  of  cubic  feet  of  air  transmitted  per  minute 
in  pipes  of  various  diameters  is  given  by  5'  =  .327  vd''^,  where 
q  =  number  of  cubic  feet  of  air  per  minute,  v  =  velocity  of  flow 
in  feet  per  second,  and  d  =  diameter  of  pipe  in  inches. 

Make  a  table  showing  the  volume  of  air  transmitted  in  pipes 
of  diameters  from  2  to  10  in.  with  a  flow  of  12  ft.  per  second. 


LOGARITHMIC  PAPER 


255 


14.  The  following  formula  is  used  for  computing  the  surface 
curvature  in  paving  streets :  y  =  -ix^,  where  x  =  horizontal  dis- 
tance in  feet  from  center  of  street,  y  =  vertical  distance  in  inches 
below  grade,  a  =  one  half  the  width  of  the  street  in  feet,  b  = 
depth  of  gutter  in  inches  below  center  of  street. 


Fig.  110 

Construct  a  graph  to  read  off  the  vertical  distances  below 
grade  at  points  2,  4,  6  ft.  •  •  •  from  the  center  of  a  street  60  ft. 
wide,  if  the  gutter  is  15  in.  below  the  center  of  the  street. 

Find  the  equation  connecting  x  and  y  when  the  following 
corresponding  values  are  given  : 


15. 


Suggestion.  Locate  the  points  and  draw  a  line  through  them,  cut- 
ting OX  at  A  and  YZ  at  B.  From  C  on  YZ  directly  above  .4  draw 
a  line  parallel  to  BA,  cutting  OF  at  X).  OD  =  3.5  =  m.  The  slope  of 
.4/>  is  2  ;  hence  the  required  equation  is  y  =  3.5  x^. 


X 

V 

2 
14 

2.5 

2i.n 

3 
31.5 

3.5 
42.9 

4 

56 

16. 


17. 


X 

y 

2 
32 

3 
108 

4 

256 

5 
500 

6 
864 

X 

y 

4 
4 

5 
4.47 

6 
4.90 

7 
5.29 

8 
5.66' 

256 


APPLIED  MATHEMATICS 


X 

y 

1.61 
220 

2.01 
230 

3.05 
250 

4.48 
.   270 

7.59 
300 

X 

y 

20 
1099 

30 

2248 

40 
3826 

50 
5717 

60 
7943 

18. 


Suggestion.  The  line  through  the  points  cuts  OF  at  2.  The  values 
of  y,  however,  suggest  that  it  should  be  read  200,  and  this  will  be 
found  to  be  correct  on  checking. 


19. 


Suggestion.  Let  the  line  through  the  points  cut  OX  at  A  and  YZ 
at  B.  From  C  on  OX  directly  below  B  draw  CD  to  XZ  parallel  to 
AB;  and  from  E  on  YZ  directly  above  A  draw  EF  to  OY  parallel 
to  AB.  FE  —  AB  —  CD  is  the  part  of  the  graph  for  values  of  x  from 
10  to  100.  To  find  m  construct  the  part  of  the  graph  for  values  of 
X  from  10  to  1. 


20. 


Find  the  law  connecting  the  two  variables  in  the  following : 

21.  In  a  test  of  cast-iron  columns  6  ft.  long,  both  ends 
rounded,  the  following  results  were  obtained,  where  d  =  diame- 
ter of  column  in  inches,  and  t  =  load  in  tons  under  which  the 
column  broke  by  bending. 


X 

y 

15 
486 

20 

589 

25 
684 

30 

772 

64 
1280 

2 
10.7 


2.5 
24.9 


3 

49.4 


3.5 

88.2 


4 
146 


22.  The  bearing  end  of  a  vertical  shaft  is  called  a  pivot.  For 
slow-moving  steel  pivots  the  following  table  of  values  is  given, 
where  d  =  diameter  of  pivot  in  inches,  and  p  =  total  vertical 
pressure  on  the  pivot  in  pounds. 


1 
816 


1.5 
1836 


2 
3265 


2.5 
5102 


3 
7347 


3.5 
10,000 


4 
13,061 


4.5 
16,530 


LOGARITHMIC  PAPER 


257 


23.  The  following  table  gives  the  absolute  temperature 
(F.)  of  air  at  different  pressures  when  it  is  compressed  without 
gain  or  loss  of  heat,  t  —  absolute  temperature  (F.),  and  2^  = 
pounds  per  square  inch. 


p 

15 

30 

45 

60 

90 

t 

530 

649 

730 

792 

892 

24.  The  following  results  were  obtained  in  a  test  in  towing 
a  canal  boat,  p  =  pull  in  pounds,  and  v  =  speed  of  boat  in  miles 
per  hour. 


76 


160 
2.43 


240 
3.18 


320 
3.60 


370 
4.03 


In  the  following  examples  find  the  law  connecting  j^  and  v. 
The  expansion  is  without  gain  or  loss  of  heat,  and  p  and  v  are 
corresponding  values  of  the  pressure  and  volume. 

25.  Steam. 


1 
100 


2 
37.7 


3 

21.3 


5 
10.4 


7 
6.48 


9 
4.54 


26.  Steam. 


V 

3 

4 

6 

8 

10 

p 

118 

90.8 

63.3 

48.9 

40 

27.  Superheated  steam. 


2 
105 


3 
61.8 


5 
52 


7 
20.7 


15 


28.  Mixture  in  cylinder  of  a  gas  engine. 


V 

2 

4 

6 

8 

10 

p 

57 

21.2 

11.8 

8.1 

5.9 

258 


APPLIED  MATHEMATICS 
Wire  Table  —  Copper  Wire 


Area  in 

Diameter  in  mils 

Resistance, 

Weight,  pounds 

circular  mils 

ohms  per  1000  ft. 

per  1000  ft. 

2,000,000 

1414 

.00519 

6044 

1,750,000 

1323 

.00593 

5289 

1,500,000 

1225 

.00692 

4533 

1,250,000 

1118 

.008.30 

3778 

1,000,000 

1000 

.010:W 

3022 

950,000 

974.7 

.01093 

2871 

o 

900,000 

948.7 

.01153 

2720 

H 

850,000 

922.0 

.01221 

2569 

p^ 

800,000 

894.4 

.01298 

2418 

< 

750,000 

866.0 

.01384 

2266 

CO 

700,000 

836.7 

.01483 

2115 

Q 

650,000 

806.2 

.01597 

1964 

600,000 

774.6 

.01730 

1813 

<l 

550,000 

741.6 

.01887 

1662 

S5 

500,000 

707.1 

.02076 

1511 

1 

450,000 

670.8 

.02;507 

1360 

M 

400,000 

632.5 

.02595 

1209 

PQ 

350,000 

591.6 

.02966 

1058 

300,000 

547.7 

.03460 

906.5 

250,000 

500.0 

.041.52 

755.5 

225,000 

474.3 

.04614 

680.0 

0000 

211,600 

460.00 

.04906 

639.33 

000 

167,805 

409.64 

.06186 

507.01 

00 

133,079 

364.80 

.07801 

402.09 

0 

105,592 

324.95 

.09831 

319.04 

1 

83,694 

289.30 

.12401 

252.88 

2 

66,373 

257.63 

.15640 

200.54 

3 

52,634 

229.42 

.19723 

159.03 

4 

41,742 

204.31 

.24869 

126.12 

5 

33,102 

181.94 

.31361 

100.01 

6 

26,251 

162.02 

.39546 

79.32 

7 

20,816 

144.28 

.49871 

62.90 

8 

16,509 

128.49 

.62881 

49.88 

9 

13,094 

114.43 

.79281 

39.56 

10 

10,381 

101.89 

1.0000 

31.37 

12 

6,529.9 

80.808 

1.5898 

19.73 

14 

4,106.8 

64.084 

2.5908 

12.41 

16 

2,582.9 

50.820 

4.0191 

7.81 

18 

1,624.3 

40.303 

6.3911 

4.91 

19 

1,288.1 

35.890 

8.2889 

3.89 

20 

1,021.5 

31.961 

10.163 

3.09 

22 

642.70 

25.347 

16.152 

1.94 

24 

404.01 

20.100 

25.695 

1.22 

28 

159.79 

12.641 

64.966 

!48 

32 

63.20 

7.950 

164.26 

.19 

36 

25.00 

5.000 

415.24 

.08 

40 

9.89 

3.144 

1049.7 

.03 

TABLES 

UNIT  EQUIVALENTS 
Pressure 

1  pound  per  square  inch       .     .     .  2.042  inches  of  mercury  at  62° 

1  pound  per  square  inch       .     .     .  2.309  feet  of  water  at  62°  F. 

1  atmosphere 14.7  pounds  per  square  inch. 

1  atmosphere 30  inches  of  mercury  at  62°  F. 

1  atmosphere 33.95  feet  of  water  at  62°  F. 

1  foot  of  water  at  62°  F 433  pound  per  square  inch. 

1  inch  of  mercury  at  62°  F.       .     .  .491  pound  per  square  inch. 

Length 

Imil 001  inch. 

1  inch 2.54  centimeters. 

1  mile 1.609  kilometers. 

1  centimeter 3937  inch. 

1  kilometer 3280.8  feet. 

Area 

1  circular  mil 7854  square  mil. 

1  square  mil 1.273  circular  mils. 

1  square  inch 645.16  square  millimeters, 

1  square  centimeter 155  square  inch. 

Volume 

1  cubic  inch 16.387  cubic  centimeters. 

1  cubic  foot 7.48  gallons  (liquid,  U.  S.). 

1  pint  (liquid,  U.  S.) 473.18  cubic  centimeters. 

1  pint  (liquid,  U.  S.) 28.875  cubic  inches. 

1  gallon  (liquid,  U.  S.)     .     .     .     .     231  cubic  inches. 

1  bushel 2150.4  cubic  inches. 

1  cubic  centimeter 061  cubic  inch. 

1  liter 61.02  cubic  inches. 

1  liter 2.113  pints  (liquid,  U.  S.). 

259 


260 


APPLIED  MATHEMATICS 


Weight 

1  ounce  (avoirdupois) 
1  ounce  (avoirdupois) .     . 
1  pound  (avoirdupois) 
1  ton  (2000  pounds)     .     . 
1  cubic  centimeter  of  water 

1  gram 

1  cubic  foot  of  water  .  , 
1  cubic  inch  of  water  .  . 
1  gallon  of  water  (liquid,  U 


S.) 


437.5  grains. 
28.35  grams. 

453.6  grams. 
907.185  kilograms. 
1  gram. 

.0353  ounce  (avoirdupois). 
62.4  pounds. 
.0361  pounds. 
8.345  pounds. 


Energy,  Work,  Heat 

1  British  thermal  unit  (B.  t.  u.)     .  1  pound  water  1°  F. 

1  British  thermal  unit      ....  778  foot  pounds. 

1  British  thermal  unit 293  watt  hour. 

1  horse  power  hour 746  watt  liours. 

1  horse  power  hour 2544.7  British  thermal  units. 

1  kilowatt  hour 3412.66  British  thermal  units. 

1  kilowatt  hour 1.341  horse  power  hours. 

Power 

1  watt 44.25  foot  pounds  per  minute. 

1  watt 0569  B.  t.  u.  per  minute. 

1  horse  power 33,000  foot  pounds  per  minute. 

1  horse  power 746  watts  per  minute. 

1  horse  power 42.41  B.  t.  u.  per  minute. 


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262  APPLIED  MATHEMATICS 

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University  of  Chicago  Press.     75  cents. 
Stone-Mi LLis.    Elementary  Plane  Geometry,  pp.  252.    1910.    Benj.  H. 

Sanborn  &  Co.     80  cents. 
Wright.   Exercises  in  Concrete  Geometry,*  pp.  84.    1906.    D.  C.  Heath 

&  Co.     30  cents. 

BoHANNAN.   Plane  Trigonometry,  pp.  374.   1904.  Allyn  &  Bacon.   $2.50. 
Playne  and  Fawdry.    Plane  Trigonometry,  pp.  176.    1907.    Arnold. 
2s.  6d. 

Miller.   Progressive  Problems  in  Physics,  pp.  218.    1909.    D.  C.  Heath 

&  Co.     60  cents. 
Sanborn.   Mechanics  Problems,  pp.  194.   John  Wiley  &  Sons.     $1.50. 
Snvdeh  and  Palmer.     One  Thousand  Problems  in  Physics,  pp.  142. 

1902.    Ginn  and  Company.     55  cents. 


264  APPLIED  MATHEMATICS 

Atkinson.    Electrical  and  Magnetic  Calculations,  pp.  310.   1908.  D.  Van 

Nostrand  Company.     $1.50. 
Hooper  and  Wells.    Electrical  Problems,  pp.  170.    Ginn  and  Company. 

$1.25. 
James  and  Sands.    Elementary  Electrical  Calculations,  pp.  224,    1905. 

Longmans,  Green  &  Co.     |1.25. 
Shei'ardson.    Electrical  Catechism,  pp.  417.    1908.    McGraw-Hill  Book 

Company.     |2.00. 
Whittaker.    Arithmetic  of  Electrical  Engineering,  pp.  159.    Whlttaker. 

25  cents. 

Behrendsen-Gotting.  LehrbuchderMathematik,  254S.  1909.  Teubner. 

M.  2.80. 
Ehrig.    Geometric  fiir  Baugewerkenschulen,  Teil  I.,  138  S.   1909.  Lelne- 

weber.     M.  2.80. 
Fenkner.    Arithmetische  Aufgaben,  Ausgabe  A.,Teil  I.,274S.    M.  2.20. 

Teil  Ila.,  114S.     M.  1.50.        Teil  116.,  218S.     M.  2.60.         Salle. 
GeigenmUller.    Hohere  Mathematik,  1.  290  S.    1907.    Polytechnische 

Buchhandlung.     M.  6. 
Mil LLER  UND  Kutnewsky.    Sammlung  von  Aufgaben  aus  der  Arithmetik, 

Trigonometric  und  Stereometrie,  Ausgabe  B.,  2ter  Teil,  312  S.    1910. 

Teubner.     M.  3. 
ScHiJLKE.  Aufgaben-Sammlung,  Teil  L,  194  S.   1906.   Teubner.    M.  2.20. 
Weill.   Sammlung  Graphischer  Aufgaben  fiir  den  Gebrauch  an  hohere 

Schulen,  64S.    1909.   J.  Boltzesche  Buchhandlung.     M.  1.80. 


I.  Four-place  Logarithms  of  Three-Figure  Numbers 

II.  The  Natural  Sines,  Cosines,  Tangents,  and  Cotan- 
gents OF  Angles  differing  by  Ten  Minutes,  and  their 
Four-place  Logarithms 


265 


266 


APPLIED  MATHEMATICS 


1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0000 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

1 

0000 

0414 

0792 

1139 

1461 

1761 

2041 

2304 

2553 

2788 

2 

3010 

3222 

3424 

3617 

3802 

3979 

4150 

4314 

4472 

4624 

3 

4771 

4914 

5051 

5185 

5315 

5441 

5563 

5682 

5798 

5911 

4 

6021 

6128 

6232 

6335 

6435 

6532 

6628 

6721 

6812 

6902 

5 

6990 

7076 

7160 

7243 

7324 

7404 

7482 

7559 

7634 

7709 

6 

7782 

7853 

7924 

7993 

8062 

8129 

8195 

8261 

8325 

8388 

7 

8451 

8513 

8573 

8633 

8692 

8751 

8808 

8865 

8921 

8976 

8 

9031 

9085 

9138 

9191 

9243 

9294 

9345 

9395 

9445 

9494 

9 

9542 

9590 

9638 

9685 

9731 

9777 

9823 

9868 

9912 

9956 

10  "■ 

0000 

0043 

0086 

0128- 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0007 

0645 

0682 

0719 

0755 

12 

0792 

0828 

08G4 

0899 

0934 

0969 

1004 

1038 

1072 

HOC 

13. 

1139 

1173 

1206 

1239' 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

IG 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227' 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

25M 

2529 

18 

2553 

2577 

2601 

2Q25 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

2C 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27  • 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

0064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

0222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6135 

6444 

6454 

6464 

6474 

6184 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

FOUR-PLACE  LOGARITHMS 


267 


60 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

61 

7076 

7081 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7620 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7601 

7612 

7619 

7027 

58 

7634 

7642 

7649 

7657 

7661 

7672 

7679 

7686 

7694 

7701 

69 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

77CT 

7774 

eo 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8011 

8048 

8066 

(A 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

66 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8236 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

60 

8388 

8396 

8401 

8407 

8414 

8120 

8426 

8432 

8439 

8145 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8188 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8513 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8615 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8901 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

90K 

80 

9031 

9036 

9012 

9047 

9053 

9068 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9;i50 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9406 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9446 

9450 

9465 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9601 

9509 

9513 

9618 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9666 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9009 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9617 

9652 

9657 

9661 

9666 

9671 

9675 

9080 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9761 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

§? 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9851 

9859 

9863 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9962 

99 

/,9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

100 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

268 


APPLIED  MATHEMATICS 


Angle 

Sines 

Cosines 

Tang 

ENTS 

COTAN 

sents 

Ang  le 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

0°00' 

.0000 

00 

1.0000 

0.0000 

.0000 

00 

00 

00 

90^00' 

10 

.0029 

7.4637 

1.0000 

0000 

.0029 

7.4637 

2.5363 

343.77 

50 

20 

.0058 

7648 

1.0000 

0000 

.0058 

7648 

2352 

171.89 

40 

30 

.0087 

9408 

1.0000 

0000 

.0087 

9409 

0591 

114.59 

30 

40 

.0116 

8.0658 

.9999 

0000 

.0116 

8.0658 

1.9342 

85.940 

20 

50 

.0145 

1627 

.9999 

0000 

.0145 

1627 

8373 

68.750 

10 

1°00' 

.0175 

8.2419 

.9998 

9.9999 

.0175 

8.2419 

1.7581 

57.290 

89O00' 

10 

.0201 

3088 

.9998 

9999 

.0204 

3089 

6911 

49.101 

50 

20 

.0233 

3668 

.9997 

9999 

.0233 

3669 

6331 

42.904 

40 

30 

.0202 

4179 

.9997 

9999 

.0262 

4181 

5819 

38.188 

30 

40 

.0291 

4037 

.9996 

9998 

.0291 

4638 

5362 

34.368 

20 

50 

.0320 

5050 

.9995 

9998 

.0320 

5053 

4947 

31.242 

10 

2°  00' 

.0349 

8.5428 

.9994 

9.9997 

.0349 

8.5431 

1.4569 

28.636 

88°  OC 

10 

.0378 

5770 

.9993 

9997 

.0378 

5779 

4221 

26.432 

50 

20 

.0407 

-6097 

.9992 

9996 

.0407 

6101 

3899 

24.542 

40 

30 

.0136 

6397 

.9990 

9996 

.0437 

6401 

3599 

22.904 

30 

40 

.0165 

6677 

.9989 

9995- 

.0466 

6682 

3318 

21.470 

20 

50 

.0494 

6940 

.9988 

9995 

.0495 

6945 

3055 

20.206 

10 

3°  00' 

.0523 

8.7188 

.9986 

9.9994 

.0524 

8.7194 

1.2806 

19.081 

87°  00' 

10 

.0552 

7423 

.9985 

9993 

.0553 

7429 

2571 

18.075 

50 

20 

.0581 

7645 

.9983 

9993 

.0582 

7652 

2348 

17.169 

40 

30 

.0610 

7857 

.9981 

9992 

.0012 

7865 

2135 

16.350 

30 

40 

.OfrlO 

8059 

.9980 

9991 

.0011 

8007 

1933 

15.605 

20 

50 

.0669 

8251 

.9978 

9990 

.0670 

8261 

1739 

14.924 

10 

4°  00' 

.0698 

8.8430 

.9976 

9.9989 

.0699 

8.8446 

1.1554 

14.301 

86°  00' 

10 

.0727 

8613 

.9974 

9989 

.0729 

8624 

1376 

13.727 

60 

20 

.0756 

8783 

.9971 

9988 

.0758 

8795 

1205 

13.197 

40 

30 

.0785 

8946 

.9969 

9987 

.0787 

8960 

1040 

12.700 

30 

40 

.0814 

9104 

.9967 

9986 

.0816 

9118 

0882 

12.251 

20 

50 

.0843 

9256 

.9964 

9985 

.0846 

9272 

0728 

11.826 

10 

5°  00' 

.0872 

8.9403 

.9902 

9.9983 

.0875 

8.9420 

1.0580 

11.430 

85°  00' 

10 

.0901 

9,545 

.99,59 

9982 

.0!)04 

9563 

0437 

11.059 

50 

20 

.0929 

9682 

.9957 

9981 

.0934 

9701 

0299 

10.712 

40 

30 

.0958 

9816 

.99.54 

9980 

.0963 

9836 

0104 

10.385 

30 

40 

.0987 

9945 

.9951 

9979 

.0992 

9966 

0034 

10.078 

20 

50 

.1016 

9.0070 

.9948 

9977 

.1022 

9.0093 

0.9907 

9.7882 

10 

6°  00' 

.1(H5 

9.0192 

.9945 

9.9976 

.1051 

9.0216 

0.9784 

9. .5144 

84°  00' 

10 

.1074 

0311 

.9942 

9975 

.1080 

0336 

9664 

9.2553 

50 

20 

.1103 

0426 

.9939 

9973 

.1110 

0453 

9547 

9.0098 

40 

30 

.1132 

0539 

.9930 

9972 

.1139 

0567 

9433 

8.7769 

30 

40 

.1161 

0G48 

.9932 

9971 

.1169 

0678 

9322 

8.5,555 

20 

50 

.1190 

0755 

.9929 

9969 

.1198 

■  0786 

9214 

8.3450 

10 

7°  00' 

.1219 

9.0859 

.9925 

9.9968 

.1228 

9.0891 

0.9109 

8.1443 

83°  00' 

10 

.1248 

0961 

.9922 

9966 

.1257 

0995 

9005 

7.9530 

50 

20 

.1276 

lOCO 

.9918 

9964 

.1287 

1096 

8904 

7.7704 

40 

30 

.1305 

,  11.57 

.9914 

9963 

.1317 

1194 

8806 

7.5958 

30 

40 

.1334 

1252 

.9911 

9961 

.1346 

1291 

8709 

7.4287 

20 

50 

.1363 

1345 

.9907 

9959 

.1376 

1385 

8615 

7.2687 

10 

"8°  00' 

.1392 

9.1436 

.9903 

9.9958 

.1405 

9.1478 

0.8522 

7.1154 

82°  00' 

10 

.1421 

1525 

.9899 

9956 

.1435 

1569 

8431 

6.9682 

50 

20 

.1449 

1612 

.9894 

9954 

.1465 

1658 

8342 

6.8269 

40 

30 

.1478 

1697 

.9890 

9952 

.1495 

1745 

8255 

6.6912 

30 

40 

.1507 

1781 

.9886 

9950 

.1524 

1831 

8169 

65606 

20 

50 

.1536 

1863 

.9881 

9948 

.1554 

1915 

8085 

6.4348 

10 

9°  00' 

.1564 

9.1943 

.9877 

9.9946 

.1584 

9.1997 

0.8003 

6.3138 

81°  00' 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

Ang  le 

Cosines 

Sines 

Cotangents 

Tangents 

Angle 

FOUR-PLACE  LOGARITHMS 


269 


Angle 

Sines 

Cosines 

Tangents 

Cotangents 

Angle 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

gooc 

.15G4 

9.1943 

.9877 

9.9946 

.1584 

9.1997 

0.8003 

6.3138 

81°  00' 

10 

.1593 

2022 

.9872 

9944 

.1614 

2078 

7922 

6.1970 

50 

20 

.1622 

2100 

.9868 

9942 

.1644 

2158 

7842 

6.0844 

40 

30 

.1050 

2176 

.9863 

9440 

.1673 

2236 

7761 

5.9758 

30 

40 

.1679 

2251 

.9858 

9938 

.1703 

2313 

7687 

5.8708 

20 

50 

.1708 

2324 

.9853 

9936 

.1733 

2389 

7611 

5.7694 

10 

10°  00' 

.1736 

9.2397 

.9848 

9.9934 

.1763 

9.2463 

0.7537 

"5.6713 

80°  00' 

10 

.1765 

2468 

.9843 

9931 

.1793 

2530 

7464 

5.5764 

50 

20 

.1794 

2538 

.9838 

9929 

.1823 

2609 

7391 

5.4845 

40 

30 

.1822 

2606 

.9833 

9927 

.1853 

2680 

7320 

5.3955 

30 

40 

.1851 

2674 

.9827 

9924 

.1883 

2750 

7250 

5.3093 

20 

60 

.1880 

2740 

.9822 

9922 

.1914 

2819 

7181 

5.2257 

10 

11°  00' 

.1908 

9.2806 

.9816 

9.9919 

.1944 

9.2887 

0.7113 

5.1446 

79°  00' 

10 

.1937 

2870 

.9811 

9917 

.1974 

2953 

7047 

5.0658 

50 

20 

.1965 

2934 

.9805 

9914 

.2004 

3020 

6980 

4.9894 

40 

30 

.1994 

2997 

.9799 

9912 

.2035 

3085 

6915 

4.9152 

30 

40 

.2022 

3058 

.9793 

9909 

.2065 

3149 

6851 

4.8430 

20 

50 

.2051 

3119 

.9787 

9907 

.2095 

3212 

6788 

4.7729 

10 

12°  00' 

.2079 

9.3179 

.9781 

9.9901 

.2120 

9.3275 

0.0725 

4.7040 

78°  00' 

10 

.2108 

3238 

.9775 

9901 

.2156 

3336 

6664 

4.6382 

50 

20 

.2136 

3296 

.9769 

9899 

.2186 

3397 

6603 

4..')736 

40 

30 

.2161 

3353 

.9763 

9896 

.2217 

3458 

6542 

4.5107 

30 

40 

.2193 

3410 

.9757 

9893 

.2247 

3517 

0183 

4.4494 

20 

50 

.2221 

3466 

.9750 

9890 

.2278 

3570 

6424 

4.3897 

10 

13°  00' 

.2250 

9.3521 

.9744 

9.9887 

.2309 

9.3034 

0.0300 

4.3315 

77°  00' 

10 

.2278 

3575 

.9737 

9884 

.2339 

3691 

0309 

4.2747 

50 

20 

.2306 

3629 

.9730 

9881 

.2370 

3748 

6252 

4.2193 

40 

30 

.2334 

3682 

.9724 

9878 

.2401 

3804 

6196 

4.1653 

30 

40 

.2363 

3734 

.9717 

9875 

.2432 

3859 

6141 

4.1126 

20 

50 

.2391 

3786 

.9710 

9872 

.2462 

3914 

6086 

4.0611 

10 

14°  00' 

.2419 

9.3837 

.9703 

9.9860 

.2493 

9.3968 

0.6032 

4.0108 

76°  00' 

10 

.2447 

3887 

.9696 

9866 

.2524 

4021 

5979 

3.9617 

50 

20 

.2476 

3937 

.9689 

9863 

.2555 

4074 

5926 

3.9136 

40 

30 

.2501 

3986 

.9681 

9859 

.2586 

4127 

5873 

3.8667 

30 

40 

.2532 

4035 

.9674 

9856 

.2617 

4178 

5822 

3.8208 

20 

50 

.2560 

4083 

.9667 

9853 

.2648 

4230 

5770 

3.7700 

10 

15°  00' 

.2588 

9.4130 

.9659 

9.9849 

.2679 

9.4281 

05719 

3.7321 

75°  00' 

10 

.2616 

4177 

.9652 

9846 

.2711 

4331 

5669 

3.6891 

50 

20 

.2644 

4223 

.9644 

9843 

.2742 

4381 

5619 

3.6170 

40 

30 

.2672 

4269 

.9636 

9839 

.2773 

4430 

5570 

3.6059 

30 

40 

.2700 

4314 

.9628 

9836 

.2805 

4479 

5521 

3.5656 

20 

50 

.2728 

4359 

.9621 

9832 

.2836 

4527 

5473 

3.5261 

10 

1C°00' 

.2756 

9.4403 

.9613 

9.9828 

.2867 

9.4575 

0.5425 

3.4874 

74°  00' 

10 

.2784 

4447 

.9605 

9825 

.2899 

4622 

5378 

3.4495 

50 

20 

.2812 

4491 

.9596 

9821 

.2931 

4669 

5331 

3.4124 

40 

30 

.2840 

4533 

.9588 

9817 

.2962 

4716 

5284 

3.3759 

30 

40 

.2868 

4576 

.9580 

9814 

.2994 

4762 

5238 

3.3402 

20 

50 

.2896 

4618 

.9572 

9810 

.3020 

4808 

5192 

3.3052 

10 

17°  00' 

.2924 

9.4659 

.9563 

9.9806 

.3057 

9.4853 

0.5147 

3.2709 

73°  00' 

10 

.2952 

4700 

.9555 

9802 

.3080 

4898 

5102 

3.2371 

50 

20 

.2979 

4741 

.9546 

9798 

.3121 

4943 

5057 

3.2041 

40 

30 

.3007 

4781 

.9537 

9794 

.3153 

4987 

5013 

3.1716 

30 

40 

.3035 

4821 

.9528 

9790 

.3185 

5031 

4909 

3.1397 

20 

50 

.3062 

4861 

.9520 

9786 

.3217 

5075 

4925 

3.1084 

10 

18°  00' 

.3090 

9.4900 

.9511 

9.9782 

.3249 

9.5118 

0.4882 

3.0777 

72°  OO' 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

Angle 

Cosines 

Sines 

Cotangents 

Tangents 

Ang  le 

270 


APPLIED  MATHEMATICS 


Angle 

Sines 

Cosines 

Tangents 

COTAN 

GENTS 

Angle 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

18°  00' 

.3090 

9.4900 

.9511 

9.9782 

.3249 

95118 

0.4882 

3.0777 

72°  00' 

10 

.3118 

4939 

.9502 

9778 

.3281 

5161 

4839 

3.0475 

50 

20 

.3145 

4977 

.9492 

9774 

.3314 

5203 

4797 

3.0178 

40 

30 

.3173 

5015 

.9483 

9770 

.3346 

5245 

4755 

2.9887 

30 

40 

.3201 

5052 

.9474 

9765 

.3378 

5287 

4713 

2.9600 

20 

50 

.3228 

5090 

.9465 

9761 

.3411 

6329 

4671 

2.9319 

10 

19°  00' 

.3256 

9.5126 

.9455 

9.9757 

.3443 

9.5370 

0.4630 

2.9042 

71°  00' 

10 

.3283 

5163 

.9446 

9752 

.3476 

5411 

4589 

2.8770 

50 

20 

.3311 

5199 

.9436 

9748 

.3508 

5451 

4549 

2.8502 

40 

30 

.3338^ 

5235 

.9426 

9743 

.3541 

5491 

4509 

2.8239 

30 

40 

.3365 

5270 

.9417 

9739 

.3574 

5531 

4469 

2.7980 

20 

50 

.3393 

5306 

.9407 

9734 

.3607 

5571 

4429 

2.7725 

10 

20°  00' 

.3420 

9.5341 

.9397 

9.9730 

.3640 

9.5611 

0.4389 

2.7475 

70°  Oy 

10 

.3448 

5375 

.9387 

9725 

.3673 

5650 

4350 

2.7228 

50 

20 

.3475 

5409 

.9377 

9721 

.3706 

5689 

4311 

2.6085 

40 

30 

.3502 

5443 

.9367 

9716 

.3739 

5727 

4273 

2.6746 

30 

40 

.3529 

5477 

.9356 

9711 

.3772 

5766 

4234 

2.0511 

20 

50 

.3557 

5510 

.9346 

9706 

.3805 

5804 

4196 

2.6279 

10 

21°  00' 

.3584 

95543 

.9336 

9.9702 

.3839 

95842 

0.4158 

2.6051 

69°  00' 

10 

.3611 

5576 

.9325 

9697 

.3872 

5879 

4121 

2.5826 

50 

20 

.3638 

5609 

.9315 

9692 

.3906 

5917 

4083 

25605 

40 

30 

.3665 

5611 

.9304 

9687 

.3939 

5954 

4046 

2.5386 

30 

40 

.3692 

5673 

.9293 

9682 

.3973 

5991 

4009 

25172 

20 

50 

.3719 

5704 

.9283 

9677 

.4006 

6028 

3972 

2.4900 

10 

22°  00' 

.3746 

9.5736 

.9272 

9.9672 

.4040 

9.6064 

0.3936 

2.4751 

68°  00' 

10 

.3773 

5767 

.9261 

9667 

.4074 

6100 

3900 

2.4545 

50 

20 

.3800 

5798 

.9250 

9661 

.4108 

6136 

3864 

2.4342 

40 

30 

.3827 

5828 

.9239 

9656 

.4142 

6172 

3828 

2.4142 

30 

40 

.3854 

5859 

.9228 

9651 

.4176 

6208 

3792 

2.3945 

20 

50 

.3881 

5889 

.9216 

9(H6 

.4210 

6243 

3757 

2.3750 

10 

23°  00' 

.3907 

95919 

.9205 

9.9640 

.4245 

9.6279 

0.3721 

2.3559 

67°  00' 

10 

.3934 

5948 

.9194 

9635 

.4279 

6314 

3686 

2.3369 

50 

20 

.3961 

5978 

.9182 

9629 

.4314 

6348 

3652 

2.3183 

40 

30 

.3987 

6007 

.9171 

9624 

.4348 

6383 

3617 

2.2998 

30 

40 

.4014 

6036 

.9159 

9618 

.4383 

W17 

3583 

2.2817 

20 

50 

.4011 

6065 

.9147 

9613 

.4417 

6452 

3548 

2.2637 

10 

24°  00' 

.4067 

9.6093 

.9135 

9.9607 

.4452 

9.6486 

0.3514 

2.2460 

66°  00' 

10 

.4094 

6121 

.9124 

9602 

.4487 

6520 

3480 

2.2286 

50 

20 

.4120 

6149 

.9112 

9596 

.4522 

6553 

3447 

2.2113 

40 

30 

.4147 

6177 

.9100 

9590 

.4557 

6587 

3413 

2.1943 

30 

40 

.4173 

6205 

.9088 

9584 

.4592 

6620 

3380 

2.1775 

20 

50 

.4200 

6232 

.9075 

9579 

.4628 

6654 

3346 

2.1609 

10 

25°  00' 

.4226 

9.6259 

.9063 

9.9573 

.4663 

9.6687 

0.3313 

2.1445 

65°  00' 

10 

.4253 

6286 

.9051 

9567 

.4699 

6720 

3280 

2.1283 

50 

20 

.4279 

6313 

.9038 

9561 

.4734 

6752 

3248 

2.1123 

40 

30 

.4305 

6340 

.9026 

9555 

.4770 

6785 

3215 

2.0965 

30 

40 

.4331 

6366 

.9013 

9549 

.4806 

6817 

3183 

2.0809 

20 

50 

.4358 

6392 

.9001 

9513 

.4841 

6850 

3150 

,2.0655 

10 

26°  00' 

.4384 

9.6418 

.8988 

9.9537 

.4877 

9.6882 

0.3118 

2.0503 

64°  00' 

10 

.4410 

6144 

.8975 

9530 

.4913 

6914 

3086 

2.0353 

50 

20 

.4436 

6170 

.8962 

9521 

.4950 

6946 

3054 

2.0204 

40 

30 

.4462 

6495 

.8949 

9518 

.4986 

6977 

3023 

2.0057 

30 

40 

.4488 

6521 

.8936 

9512 

5022 

7009 

2991 

1.9912 

20 

50 

.4514 

6546 

.8923 

9505 

5059 

7040 

2960 

1.9768 

10 

27°  00' 

.4540 

9.6570 

.8910 

9.9499 

5095 

9.7072 

0.2928 

1.9626 

63°  00' 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

Angle 

Cosines 

Sines 

Cotangents 

Tangents 

Angle 

FOUR-PLACE  LOGARITHMS 


271 


Ang  le 

Sines 

Cosines 

Tangents 

Cotangents 

Angle 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

27°  00' 

.4540 

9.6570 

.8910 

9.9499 

5095 

9.7072 

0.2928 

1.9626 

63°  OO' 

10 

.4506 

6595 

.8897 

9492 

5132 

7103 

2897 

1.948(i 

50 

20 

.4592 

6620 

.8884 

9486 

5169 

7134 

2866 

1.9347 

40 

30 

.4617 

6644 

.8870 

9479 

5206 

7165 

28;J5 

1.9210 

30 

40 

.4613 

6668 

.8857 

9473 

.5213 

7196 

2801 

1.9074 

20 

50 

.4669 

6692 

.8843 

9466 

5280 

7226 

2774 

1.8940 

10 

28°  00' 

.4695 

9.6716 

.8829 

9.9459 

5317 

9.7257 

0.2743 

1.8807 

62°  00' 

10 

.4720 

6740 

.8816 

9453 

5354 

7287 

2713 

1.8676 

50 

20 

.4746 

6763 

.8802 

9446 

5392 

7317 

2683 

1.8546 

40 

30 

.4772 

6787 

.8788 

9439 

5430 

7348 

2652 

1.8418 

30 

40 

.4797 

6810 

.8774 

9432 

5467 

7378 

2622 

1.8201 

20 

50 

.4823 

6833 

.8760 

9425 

5505 

7408 

2592 

1.8165 

10 

29O00' 

.4848 

9.6850 

.8746 

9.9418 

.5543 

9.7438 

0.2562 

1.8010 

61°  00' 

10 

.4874 

6878 

.8732 

9411 

5581 

7467 

2533 

1.7917 

50 

20 

.4899 

6901 

.8718 

9404 

5619 

7497 

2503 

1.7790 

40 

30 

.4924 

6023 

.8704 

9397 

5658 

7526 

2474 

1.7675 

30 

40 

.4950 

6046 

.8689 

9390 

5696 

7556 

2144 

1.7556 

20 

50 

.4975 

6968 

.8675 

9383 

5735 

7585 

2415 

1.7437 

10 

30°  00' 

.5000 

9.6990 

.8660 

9.9375 

.5774 

9.7614 

0.2386 

1.7321 

60°  00' 

10 

.5025 

7012 

.8646 

9368 

5812 

7644 

2356 

1.7205 

50 

20 

.5050 

7033 

.8631 

9361 

.5851 

7673 

2327 

1.7090 

40 

30 

.5075 

7055 

.8616 

9353 

5890 

7701 

2299 

1.6977 

30 

40 

.5100 

7076 

.8601 

9346 

5930 

7730 

2270 

1.6864 

20 

50 

.5125 

7097 

.8587 

9338 

5969 

7759 

2241 

1.6753 

10 

31°  00' 

.5150 

9.7118 

.8572 

9.9331 

.6009 

9.7788 

0.2212 

1.6643 

59°  00' 

10 

.5175 

7139 

.8557 

9323 

.6048 

7816 

2184 

1.0534 

50 

20 

.5200 

7160 

.8542 

9315 

.6088 

7845 

2155 

1.0126 

40 

30 

J5225 

7181 

.8526 

9308 

.6128 

7873 

2127 

1.6319 

30 

40 

.5250 

7201 

.8511 

9300 

.6168 

7902 

2098 

1.6212 

20 

50 

.5275 

7222 

.8496 

9292 

.6208 

7930 

2070 

1.6107 

10 

32°  00' 

.5299 

9.7242 

.8480 

9.9284 

.6249 

9.7958 

0.2M2 

1.6003 

58°  00' 

10 

.5324 

7262 

.8405 

9276 

.6289 

7986 

2014 

15900 

50 

20 

.5348 

7282 

.8450 

9268 

.0330 

8014 

1986 

15798 

40 

30 

.5373 

7302 

.8434 

9260 

.6371 

SM2 

1958 

15697 

30 

40 

.5398 

7322 

.8418 

9252 

.6112 

8070 

1930 

15597 

20 

50 

.5422 

7342 

.8403 

9244 

.6453 

8097 

1903 

1.5497 

10 

33°  00' 

5446 

9.7361 

.8387 

9.9236 

.6494 

9.8125 

0.1875 

15399 

57°  00' 

10 

.5471 

7380 

.8371 

9228 

.6536 

8153 

1847 

15301 

50 

20 

.5495 

7400 

.8355 

9219 

.6577 

8180 

1820 

1.5204 

40 

30 

.5519 

7419 

.8339 

9211 

.6619 

8208 

1792 

15108 

30 

40 

.5544 

7438 

.8323 

9203 

.6661 

8235 

1765 

15013 

20 

50 

.5568 

7457 

.8307 

9194 

.6703 

8263 

1737 

1.4919 

10 

34°  00' 

.5592 

9.7476 

.8290 

9.9186 

.6745 

9.8290 

0.1710 

1.4826 

66°  00' 

10 

.5616 

7494 

.8274 

9177 

.6787 

8317 

1683 

1.4733 

50 

20 

.5610 

7513 

.8258 

9169 

.6830 

8344 

1656 

1.4641 

40 

30 

.5664 

7531 

.8241 

9160 

.6873 

8371 

1629 

1.4550 

30 

40 

.5688 

7550 

.8225 

9151 

.6016 

8398 

1602 

1.4460 

20 

50 

5712 

7568 

.8208 

9142 

.6959 

8425 

1575 

1.4370 

10 

35°  00' 

.5736 

9.7586 

.8192 

9.9134 

.7002 

9.8452 

0.1548 

1.4281 

65°  00' 

10 

.5760 

7604 

.8175 

9125 

.7046 

8479 

1521 

1.4193 

50 

20 

5783 

7622 

.8158 

9116 

.7089 

8506 

1494 

1.4106 

40 

30 

5807 

7640 

.8141 

9107 

.7133 

8533 

1467 

1.4019 

30 

40 

5831 

7657 

.8124 

9098 

.7177 

8559 

1441 

1.3934 

20 

50 

5854 

7675 

.8107 

9089 

.7221 

8586 

1414 

1.3848 

10 

36°  00' 

5878 

9.7692 

.8090 

9.9080 

.7265 

9.8613 

0.1387 

1.3764 

MO  00' 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

Angle 

Cosines 

Sines 

Cotangents 

Tangents 

Angle 

272 


APPLIED  MATHEMATICS 


Angle 

Sines 

Cosines 

Tangents 

Cotangents 

Angle 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

36°  00' 

.5878 

9.7692 

.8000 

9.9080 

.7265 

9.8613 

0.1387 

1.3764 

54°  00' 

10 

5001 

7710 

.8073 

9070 

.7310 

8639 

1361 

1.3680 

50 

20 

.5925 

7727 

.8056 

9061 

.7a55 

8666 

1334 

1.3597 

40 

30 

.5948 

7744 

.8039 

9052 

.7400 

8692 

1308 

1.3514 

30 

40 

.5972 

7761 

.8021 

9042 

.7445 

8718 

1282 

1.3432 

20 

50 

.5995 

7778 

.8004 

9033 

.7490 

8745 

1255 

1.3351 

10 

37°  00' 

.6018 

9.7795 

.7986 

9.9023 

.7536 

9.8771 

0.1229 

1.3270 

53°  OP' 

10 

.6011 

7811 

.7969 

9014 

.7581 

8797 

1203 

1.3190 

50 

20 

.6065 

7828 

.7951 

9004 

.7627 

8824 

1176 

1.3111 

40 

30 

.6088 

7844 

.7034 

8905 

.7673 

8850 

1150 

1.3032 

30 

40 

.6111 

7861 

.7016 

8985 

.7720 

8876 

1124 

1.2954 

20 

50 

.6134 

7877 

.7898 

8075 

.7766 

8902 

1008 

1.2876 

10 

38°  00' 

.6157 

9.7893 

.7880 

9.8965 

.7813 

9.8928 

0.1072 

1.2799 

52°  00' 

10 

.6180 

7910 

.7862 

8955 

.7860 

8954 

1046 

1.2723 

50 

20 

.6202 

7026 

.7844 

8945 

.7907 

8980 

1020 

1.2647 

40 

30 

.6225 

7041 

.7826 

8935 

.7954 

9006 

0994 

1.2572 

30 

40 

.6248 

70,57 

.7808 

8925 

.8002 

0032 

0068 

1.2407 

20 

50 

.6271 

7973 

.7700 

8915 

.8050 

9058 

0042 

1.2423 

10 

39°  00' 

.6293 

9.7989 

.7771 

9.8905 

.8098 

9.9084 

0.0016 

1.2349 

51°  00' 

10 

.6316 

8004 

.7753 

8895 

.8146 

9110 

0800 

1.2276 

50 

20 

.6338 

8020 

.7735 

8884 

.8105 

9135 

0865 

1.2203 

40 

30 

.6361 

8035 

.7716 

8874 

.8243 

9161 

0839 

1.2131 

30 

40 

.6383 

8050 

.7698 

8864 

.8292 

9187 

0813 

1.2059 

20 

60 

.&m 

8066 

.7679 

8853 

.8342 

9212 

0788 

1.1988 

10 

40°  00' 

.6428 

9.8081 

.7660 

9.8843 

.8391 

9.9238 

0.0762 

1.1918 

50°  00' 

10 

.(H50 

8096 

.7642 

8832 

.8441 

9264 

0736 

1.1847 

50 

20 

.6472 

8111 

.7623 

8821 

.8491 

0289 

0711 

1.1778 

40 

30 

.6494 

8125 

.7604 

8810 

.8541 

9315 

0685 

1.1708 

30 

40 

.6517 

8140 

.7585 

8800 

.8591 

9341 

0650 

1.1640 

20 

50 

.6539 

8155 

.7566 

8789 

.8642 

9366 

0634 

1.1571 

10 

41°  00' 

.6561 

9.8169 

.7547 

9.8778 

.8693 

9.9392 

0.0608 

1.1504 

49°  00' 

10 

.6583 

8184 

.7528 

8767 

.8744 

9417 

0583 

1.1436 

50 

20 

.6604 

8198 

.7509 

8756 

.8796 

9443 

0557 

1.1369 

40 

30 

.6626 

8213 

.7490 

8745 

.8847 

9468 

0532 

1.1303 

30 

40 

.6648 

8227 

.7470 

8733 

.8899 

9494 

0506 

1.1237 

20 

50 

.6670 

8241 

.7451 

8722 

.8952 

9519 

W81 

1.1171 

10 

42°  00' 

.6691 

9.8255 

.7431 

9.8711 

.9004 

9.9544 

0.0456 

1.1106 

48°  00' 

10 

.6713 

8269 

.7412 

8699 

.9057 

9570 

0430 

1.1041 

50 

20 

.6734 

8283 

.7392 

8688 

.9110 

9595 

0405 

1.0977 

40 

30 

.6756 

8297 

.7373 

8676 

.9163 

9621 

0370 

1.0913 

30 

40 

.6777 

8311 

.7353 

8665 

.9217 

9646 

0354 

1.0850 

20 

50 

.6799 

8324 

.7333 

8663 

.9271 

9671 

0329 

1.0786 

10 

43°  00' 

.6820 

9.8338 

.7314 

9.8641 

.9325 

9.9697 

0.0303 

1.0724 

47°  00' 

10 

.6841 

8351 

.7294 

8629 

.9380 

9722 

0278 

1.0661 

50 

20 

.6862 

8365 

.7274 

8618 

.0435 

9747 

0253 

1.0599 

40 

30 

.6884 

8378 

.7254 

8606 

.0400 

9772 

0228 

1.0538 

30 

40 

.6905 

8391 

.7234 

8594 

.0545 

9798 

0202 

1.0477 

20 

50 

.6926 

8405 

.7214 

8582 

.9601 

9823 

0177 

1.0416 

10 

44°  00' 

.6047 

9.8418 

.7193 

9.8569 

.0657 

9.9848 

0.0152 

1.0355 

46°  Oy 

10 

.6067 

8431 

.7173 

8557 

.9713 

9874 

0126 

1.0295 

50 

20 

.6988 

QW/\^ 

.7153 

8545 

.9770 

9899 

0101 

1.0235 

40 

30 

.7009 
.7030 

8457 

.7133 

8532 

.9827 

9924 

0076 

1.0176 

30 

40 

8469 

.7112 

8520 

.9884 

9040 

0051 

1.0117 

20 

50 

.7050 

8482 

.7092 

8507 

.9942 

9075 

0025 

1.0068 

10 

45°  00' 

.7071 

9.8495 

.7071 

9.8405 

1.0000 

0.0000 

0.0000 

1.0000 

45°  00' 

Nat. 

Log. 

Nat. 

Log. 

Nat. 

Log. 

Log. 

Nat. 

Angle 

Cosines 

Sines 

Cotangents 

Tangents 

Angle 

INDEX 


Algebra,  geometrical  exercises  for, 

153 
Ammeter,  217 
Ampere,  214 
Angle  functions,  134 
Angles,  54,  134 
Approximate  number,  2,  120 
Archimedes,  principle  of,  47 

Beams,  36 

Brake,  Prony,  21 ;  friction,  21 

British  thermal  unit,  202 

Calipers,  vernier,  9;  micrometer,  12 

Calorie,  202 

Characteristic,  121 

Cosine,  135 

Cosines,  law  of,  146 

Crane,  148 

Density,  42 

Digit,  2 

Division,  of  approximate  numbers, 

5;  by  logarithms,  124;  by  slide 

rule,  129 
Dynamos,  236;  efficiency  of,  237, 

240 


Electromotive  force,  212 
Equations,  graphical  solution  of,  86 
Errors,  1 

Field  magnets,  236 
Foot  pound, 16 
Fulcrum,  28 
Function,  92 
Functionality,  91 

Geometry,  algebraic  applications, 
52, 97, 153  ;  exercises  in  solid,  nu- 
merical, 177;  graphical,  186 ;  alge- 
braic, 190 

Graphs,  65,  223 

Gravity,  42 

Heat,  195;  linear  expansion,  199; 
measurement  of,  202 ;  mechanical 
equivalent  of,  202  ;  specific,  204 ; 
latent,  209 ;  generated  by  an  elec- 
tric current,  231 

Horse  power,  17 

Inequality  of  numbers,  92 
Joule,  202 


Efficiency,  23,  237,  240 

Electricity,  212  ;  units,  213  ;  work 
and  power,  227;  generation  of 
heat,  231;  wiring  for  light  and 
power,  233 ;  dynamos  and  motors, 
236 


Kilowatt,  227 
Kilowatt  hour,  227 

Latent  heat,  209 
Leverage,  28 
Levers,  27 


273 


274 


APPLIED  MATHEMATICS 


Logarithmic  paper,  243 
Logaritlims,  120 


Proportion,  110 
Protractor,  54 


Ratio,  109 

"Ready  reckoner,"  69,  243 


Mantissa,  120 
Mass,  42 

Maximum  and  minimum  values,  93 
Measurements,  4 
Mechanical  advantage,  28 
Melting  points,  198 
Mil,  233 
Mil  foot,  233 

Motors,  236 ;  efficiency  of,  237,  240      Slide  rule,  128 
Multiple  circuit,  221  Squared  paper,  use  of,  65 

Multiplication,     of     approximate 
numbers,  2  ;  by  logarithms,  124 ;      Tangent,  135 


Scale,  drawing  to,  52 
Series  circuit,  210 
Significant  figures,  2 
Sine,  135 
Sines,  law  of,  144 


by  slide  rule,  129 

Numbers,  exact,  2 ;  approximate,  2, 
120 ;  scale,  91 

Ohm,  213 
Ohm's  law,  214 

Parallel  circuit,  221 
Parallel  lines,  59 
Parallelogram,  59 
Perpendicular,  55 
Power,  17,  226 
Prony  brake,  21 


Thermometers,  195 
Triangle,    of    reference,    134 ;    of 
forces,  147 

Variables,  62 

Variation,  164;  inverse,  166;  joint, 

167 
Volt,  213 
Voltmeter,  217 

Watt,  226 
Watt  minute,  226 
Weight,  42 
Work,  16,  226 


ANNOUNCEMENTS 


ELEMENTS   OF 
APPLIED   MATHEMATICS 

By  Herbert  E.  Cobb,  Professor  of  Mathematics,  Lewis  Institute,  Chicago,  111. 

i2mo,  cloth,  viii  +  274  pages,  illustrated,  ^i.oo 

This  textbook  for  high  schools  and  manual-training  schools  is  an 
attempt  to  relate  arithmetic,  algebra,  geometry,  and  trigonometry  closely 
to  one  another  and  to  connect  all  the  mathematics  with  the  work  in  the 
shops  and  laboratories.  It  replaces  the  formal,  abstract,  and  purely 
theoretical  portions  of  algebra  and  geometry  with  problems  based  on 
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cises in  the  mathematics  classroom,  where  the  pupil  by  measuring  and 
weighing  secures  his  own  data  for  numerical  computations  and  geo- 
metrical constructions.  Arithmetic  is  used  to  check  algebraic  results, 
and  algebra  is  made  a  valuable  asset  in  working  out  geometrical  prob- 
lems. The  problems  deal  with  various  phases  of  real  life,  and  in  solving 
them  the  pupil  finds  use  for  all  his  mathematics,  his  physics,  and  his 
practical  knowledge.  The  book  can  be  profitably  used  in  conjunction 
with  more  formal  texts,  if  desirable. 


VOCATIONAL   ALGEBRA 

By  George  Wentworth  and  David  Eimsene  Smith 

i2nio,  cloth,  88  pages,  illustrated,  50  cents 

"  Vocational  Algebra "  is  for  the  boy  in  the  manual-training  school 
or  the  evening  technical  class  who  is  not  going  through  high  school 
and  has  no  thought  of  higher  technical  training.  It  presents  the  sim- 
ple algebraic  conceptions  that  have  a  vocational  significance  —  the 
meaning  of  algebraic  formulas,  the  solving  of  simple  equations,  the  use 
of  the  negative,  and  the  fundamental  operations.  Any  one  who  has 
mastered  the  book  will  be  able  to  understand  and  use  the  algebra  of 
trade  journals,  artisans'  manuals,  and  handbooks  of  business.  "  Voca- 
tional Algebra  "  may,  in  many  schools,  be  suitably  and  profitably  intro- 
duced in  the  eighth  grade. 

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SHOP  PROBLEMS  IN   MATHE- 
MATICS 

By  William  E.  Brkckenridge,  Chairman  of  the  Department  of  Mathematics, 

Samuel  F.  Mersereau,  Chairman  of  the  Department  of  Woodworking, 

and  Charles  F.  Moore,  Chairman  of  the  Department  of  Metal 

Working  in  Stuyvesant  High  School,  New  York  City 

Answer  Book  furnished  on  application 


Cloth,  izmo,  z8o  pages,  illustrated,  $i.oo 


THIS  book  aims  to  give  a  thorough  training  in  the  mathe- 
matical operations  that  are  useful  in  shop  practice,  e.g.  in 
Carpentry,  Pattern  Making  and  Foundry  Work,  Forging,  and 
Machine  Work,  and  at  the  same  time  to  impart  to  the  student 
much  information  in  regard  to  shops  and  shop  materials.  The 
mathematical  scope  varies  from  additiori  of  fractions  to  natural 
trigonometric  functions.  The  problems  are  practical  applica- 
tions of  the  processes  of  mathematics  to  the  regular  work  of 
the  shop.  They  are  graded  from  simple  work  in  board  measure 
to  the  more  difficult  exercises  of  the  machine  shop.  Through 
them  students  may  obtain  a  double  drill  which  will  strengthen 
their  mathematical  ability  and  facilitate  their  shop  work. 

All  problems  are  based  on  actual  experience.  The  slide  rule  is 
treated  at  length.  Short  methods  and  checks  are  emphasized. 
Clear  explanations  of  the  mechanical  terms  common  to  shop 
work  and  illustrations  of  the  machinery  and  tools  referred  to  in 
the  text  make  the  book  an  easy  one  for  both  student  and 
teacher  to  handle. 

It  should  be  useful  in  any  school,  elementary  or  advanced, 
where  there  are  shops,  as  a  review  for  supplementary  work  or 
as  a  textbook  either  in  mathematics  or  shop  work. 

132J4 

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BETZ  AND  WEBB 

PLANE  GEOMETRY 


i2mo,  cloth,  332  pages,  $1.00 


THIS  is  a  new  presentation  of  geometry  along  psycho- 
logical as  well  as  logical  lines.  It  embodies  the  latest 
developments  in  geometry  teaching,  retaining  at  the  same  time 
all  that  was  best  in  the  old  geometries.  It  is  the  outgrowth  of 
the  extended  experience  of  two  high-school  teachers  of  note, 
and  is  a  fresh,  sane,  teachable  textbook  that  will  be  welcomed 
by  teachers  the  country  over  who  have  been  waiting  for  just 
such  a  presentation  of  the  subject. 
Some  of  the  features : 

1.  A  preliminary  course  precedes  the  demonstrative  course,  vitaliz- 
ing definitions  by  abundant  illustration  and  discussion,  cultivating"  skill 
in  the  use  of  ruler  and  compass  through  interesting  drawing  exercises, 
and  presenting  exercises  requiring  simple  reasoning  and  inference. 

2.  The  topical  plan  is  followed.  Difficult  topics  are  approached  by 
means  of  a  preliminary  discussion. 

3.  Hypothetical  figures  are  avoided. 

4.  Area  precedes  similarity. 

5.  The  incommensurable  case  is  made  unnecessary. 

6.  The  theory  of  limits  is  made  optional.  It  is  preceded  by  an  alter- 
native informal  discussion. 

7.  The  different  types  of  exercises  —  constructions,  computations, 
and  original  theorems  —  receive  approximately  equal  attention. 

8.  The  applied  problems  are  numerous  but  not  excessive  in  number. 

The  aim  throughout  is  to  make  the  pupil  independent 
of  the  textbook 

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HAWKES,  LUBY,  AND  TOUTON'S 
ALGEBRAS 

By  Herbert  E.  Hawkks,  Professor  of  Mathematics  in  Columbia  University, 
William  A.  Luby,  Head  of  the  Department  of  Mathematics,  Cen- 
tral High  School,  Kansas  City,  Mo.,  and  Frank  C.  Touton, 
Principal  of  Central  High  School,  St.  Joseph,  Mo. 

FIRST    COURSE    IN   ALGEBRA    lamo,  cloth,  vii  +  334  pages,  illus- 
trated, ^i.oo. 

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illustrated,  75  cents. 

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illustrated,  ^1.25. 

THE  Hawkes,  Luby,  and  Touton  Algebras  offer  a  fresh  treat- 
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teaching  algebra  with  what  is  most  valuable  in  recent  developments. 
The  authors'  unhackneyed  and  vital  manner  of  presenting  the  sub- 
ject makes  a  sure  appeal  to  the  interest  of  the  student,  while  their 
genuine  respect  for  mathematical  thoroughness  and  accuracy  gives 
the  teacher  confidence  in  their  work. 

Arnong  the  distinctive  features  of  these  algebras  are  the  correla- 
tion of  algebra  with  arithmetic,  geometry,  and  physics;  the  liberal 
use  of  illustrative  material,  such  as  brief  biographical  sketches  of  the 
mathematicians  who  have  contributed  materially  to  the  science ;  early 
and  extended  work  with  graphs ;  and  the  introduction  of  numerous 
"  thinkable  "  problems.  Prominence  is  given  the  equation  through- 
out, and  the  habit  of  checking  results  is  constantly  encouraged. 
Thoroughness  is  assured  by  frequent  short  reviews. 

The  aim  has  been  to  treat  in  a  clear,  practical,  and  attractive  man- 
ner those  topics  selected  as  necessary  for  the  best  secondary  schools. 
The  authors  have  sought  to  prepare  a  text  that  will  lead  the  student 
to  think  clearly  as  well  as  to  acquire  the  necessary  facility  on  the 
technical  side  of  algebra.  The  books  offer  a  course  readily  adapt- 
able to  the  varying  conditions  in  different  schools  —  the  "  Complete 
School  Algebra  "  comprising  a  one-book  course  with  material  suffi- 
cient for  at  least  one  and  one-half  year's  work,  and  the  "  First  Course" 
and  "  Second  Course "  providing  the  same  material,  but  slightly 
expanded,  in  a  two-book  course. 


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